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Symmetric Cyclotomic Hecke Algebras
205, 275]293 Ž1998. JA977339
JOURNAL OF ALGEBRA ARTICLE NO.
Symmetric Cyclotomic Hecke AlgebrasU Gunter Malle† IWR, Im Neuenheimer Feld 368, Uni¨ ersitat ¨ Heidelberg, D-69120, Heidelberg, Germany
and Andrew Mathas ‡ School of Mathematics F07, Uni¨ ersity of Sydney, Sydney, NSW 2006, Australia Communicated by Michel Broue´ Received September 30, 1997
In this paper we prove that the generic cyclotomic Hecke algebras for imprimitive complex reflection groups are symmetric over any ring containing inverses of the parameters. For this we show that the determinant of the Gram matrix of a certain canonical symmetrizing form introduced by K. Bremke and G. Malle Ž Indag. Math. 8 Ž1997., 453]469. is a unit in any such ring. On the way we show that the Ariki]Koike bases of these algebras are also quasi-symmetric. Q 1998 Academic Press
0. INTRODUCTION AND STATEMENT OF THE MAIN RESULT Work of Broue, ´ Michel, and the first author on l-blocks in finite groups of Lie type suggests that the decomposition of certain Lusztig induced characters should be governed by suitable complex reflection groups. This led Broue ´ and the first author in w4x to introduce the concept of the cyclotomic Hecke algebra H ŽW, u. associated to a complex reflection U
The first author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft. The second author thanks the Universitat ¨ Heidelberg for its hospitality during the writing of this paper and also the Sonderforschungsbereich 343 at the Universitat ¨ Bielefeld for financial support. † E-mail: [email protected]. ‡ E-mail: [email protected]. 275 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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group W. Conjecturally specializations of this algebra should occur as endomorphism algebras of Lusztig induced characters, thus explaining the aforementioned observation Žsee w4x.. This general conjecture has been proved only in a small number of cases. Nevertheless, several structural and numerical consequences of this conjecture have been verified. One of the properties of cyclotomic algebras implied by the general conjecture is that H should carry a canonical symmetrizing form making it into a symmetric algebra over the ground ring A in which just the parameters are inverted. Such a canonical symmetrizing form over the field of fractions of A was constructed in w3x for all but finitely many of the cyclotomic algebras associated to irreducible complex reflection groups. In this paper we prove that pairs of dual bases with respect to this form are defined over A: THEOREM. The generic cyclotomic algebra H s H ŽW, u. of an irreducible imprimiti¨ e complex reflection group W is symmetric o¨ er Zwu, uy1 x. The proof of this result will be given in the subsequent sections. There we will also give more precise statements about properties of the corresponding symmetrizing form. The assertion of the main theorem is well known in the case of Iwahori]Hecke algebras, i.e., for cyclotomic algebras for real reflection groups. In that case a dual basis can immediately be written in terms of the standard basis of H and the assertion follows easily. It turns out that the situation for the complex case is more subtle. We would like to point out that it is still open whether the cyclotomic algebras for all of the finitely many primitive complex reflection groups enjoy the properties stated in the theorem.
1. THE SYMMETRIZING FORM ON H ŽWn , u. In this section we recall the definition of the cyclotomic Hecke algebra H ŽWn , u. and the construction of a symmetrizing form on H ŽWn , u. m QŽu. from w3x. Let V be the n-dimensional unitary space V s ² e1 , . . . , e n : with the standard unitary inner product. Fix r ) 1, and let Wn s GŽ r, 1, n. ; GLŽ V . be the imprimitive complex reflection group on V generated by the reflections t 1 , . . . , t ny1 of order 2 with roots e2 y e1 , . . . , e n y e ny1 , respectively, and a Žcomplex. reflection t 0 of order r with root e1. On these
277
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
generators, Wn has a presentation given by the diagram BnŽ r . :
r " " t t 0
1
???
" t 2
Ž 1.1.
".
t ny1
The elements t 1 , . . . , t ny1 generate a subgroup of Wn isomorphic to the symmetric group S n , and form a set of standard generators of this Coxeter group. We will consider S n as a subgroup of Wn via this embedding. Let u s Ž u1 , . . . , u r , q . be indeterminates and A [ Zwu, uy1 x, where y1 y1 . we write uy1 [ Ž uy1 . The generic cyclotomic algebra H [ 1 , . . . , ur , q Ž . H Wn , u attached to Wn is the free unital associative A-algebra on generators T0 , T1 , . . . , Tny1 , subject to the braid relations implied by the diagram Ž1.1. and the deformed order relations
Ž T0 y u1 . ??? Ž T0 y u r . s 0 s Ž Ti y q . Ž Ti q 1 . ,
1FiFny1
Žsee w4, 4.1x.. Under the specialization u k ¬ zr k
Ž1 F k F r . ,
q ¬ 1,
Ž 1.2.
where zr [ expŽ2p irr ., the cyclotomic algebra specializes to the group algebra of Wn over Zw zr x. Further, it is known that H is a free A-algebra of rank < Wn < Žsee w2x.. Thus the subalgebra generated by T1 , . . . , Tny1 can and will be identified with the generic Iwahori]Hecke algebra H Ž S n , q . of type A ny 1. For x s Ž x 1 , . . . , x l ., x i g t 0 , t 1 , . . . , t ny14 , a reduced expression of an element x s x 1 ??? x l in Wn , let Tx [ Tx 1 ??? Tx l . Note that it is not in general true that two reduced expressions x, x X of the same element x g Wn give rise to identical elements Tx , Tx X in H . Nevertheless, this property does hold if x g S n F Wn . Let X s x N x g W 4 be an arbitrary set of reduced expressions for the elements of W. Then it is known that the set BX [ Tx N x g X 4 is an A-basis of H Žsee, for example, w3, Proposition 2.4x.. Any such basis will be called a reduced basis of H . Choose one such reduced basis BX . Let f H : H ª A be the linear form defined on BX by f H : H ª A,
f H Ž Tx . s
½
1 0
if x s 1, otherwise.
Ž 1.3.
It was shown in w3, 2Bx that the linear form f H does not depend on the particular reduced expressions chosen in the definition of the reduced basis BX . Hence f H is defined unambiguously on H . Moreover, it was
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proved in w3, Proposition 2.7x that f H is a trace form. Thus the bilinear form ² , : : H H ª A,
Ž h1 , h 2 . ¬ f H Ž h1 h 2 .
Ž 1.4.
is symmetric. Clearly the specialization of ² , : to the group algebra of Wn is the standard symmetrizing form which sends Ž w 1 , w 2 . to the coefficient of 1 in the product w 1w 2 . Since the algebra HK [ H ŽWn , u. mA K over K s QuotŽ A. is split semisimple w2x, this implies: THEOREM 1.5 w3, Theorem 2.8x. The linear form f H defines a non-degenerate symmetric form ² h1 , h 2 : [ f H Ž h1 h 2 .
for h1 , h 2 g HK
on HK . This satisfies f H Ž Tx . s 0
for all reduced expressions x of non-identity elements of Wn .
Ž 1.6. We will show that f H actually defines a non-degenerate symmetric form on H over A. This seems to be important for a further study of this algebra. Let O be a domain and H an unital associative O-algebra. Recall from w4, 1Cx that an O-basis B of H with 1 g B is called quasi-symmetric if the linear form f : H ª O defined by setting f Ž b . s d 1 b for b g B is a trace form and via Ž1.4. gives rise to a non-degenerate symmetric form on H . We then also say that B is quasi-symmetric with respect to f. In this terminology, the above theorem can be rephrased to saying that any reduced basis of H ŽWn , u.K is quasi-symmetric with respect to the form f H defined above.
2. THE ARIKI]KOIKE BASIS OF H IS QUASI-SYMMETRIC It will be convenient to introduce a further quasi-symmetric basis of H s H ŽWn , u.. For this let L1 [ T0 and inductively L i [ qy1 Tiy1 L iy1Tiy1 for 2 F i F n. It was shown by Ariki and Koike w2, Theorem 3.10x that B [ Lb11 ??? Lbnn Tw N 0 F bi F r y 1, w g S n 4
Ž 2.1.
is a basis of H ŽWn , u.. Here Tw denotes the standard basis element in the Iwahori]Hecke subalgebra H Ž S n , q . of H ŽWn , u.. Note that B is nonreduced if r ) 1 and n ) 1. We can now prove the analogue of Ž1.6. for this basis.
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
279
PROPOSITION 2.2. The Ariki]Koike basis B is quasi-symmetric with respect to f H , i.e., it satisfies fH Ž b . s 0
for all 1 / b g B.
Ž 2.3.
Proof. We compare B to a particular reduced basis BX of H and show that any 1 / b g B is a linear combination of non-identity basis elements from BX . The assertion then follows from Theorem 1.5. To describe BX we introduce the elements Lk , a [
½
Tky 1 ??? T1T0a 1
for a ) 0, for a s 0,
for 1 F k F n, 0 F a F r y 1. According to w3, Lemma 1.5x the set BX [ L1, a1 ??? L n , a nTw N 0 F a i F r y 1, w g S n 4 is a reduced basis of H Žthe corresponding set X of reduced words is clear.. Furthermore, the following relations are proved in w3, Lemma 2.3x: LEMMA 2.4. We ha¨ e
Ž a.
¡L
Ti L k , a s
~L
if i ) k or a s 0, if i s k , a / 0,
k , aTi kq 1, a
qL ky 1, a q Ž q y 1 . L k , a k , aTiq1
¢L
if i s k y 1, a / 0, if i - k y 1, a / 0,
for 1 F i F n y 1, 1 F k F n.
Ž b. L k , a L kym , c s L kymy1, c L k , aT1 c
q Ž q y 1.
Ý Ž Lky my1, aqcyi Lk , i y Lkymy1, i Lk , aqcyi . is1
for a, c / 0, k y m ) 1. LEMMA 2.5.
Let i G 1.
Ža. Let 1 F k F r y 1. Then Lki is equal to an A-linear combination of basis elements L1, c1 ??? L i, c i Tw g BX with 0 F c j F r y 1, 1 F c i F k, w g ² t 1 , . . . , t iy1 :.
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MALLE AND MATHAS
Žb. Let 1 F bi F r y 1. Then Lb11 ??? Lbi i is equal to an A-linear combination of basis elements L1, c1 ??? L i, c i Tw g BX with 0 F c j F r y 1, c i / 0, w g ² t 1 , . . . , t iy1 :. Proof. According to our definitions we have Lki s Ž q 1y i L i, 1T1 ??? Tiy1 . k . Thus assertion Ža. is certainly correct if k s 1. Now assume that Ža. has been proved for all exponents strictly smaller than k. Hence Lki s 1 Lky L i is a linear combination of terms L1, c1 ??? L i, c i T¨ L i, 1Tw with 1 F c i i F k y 1 and ¨ , w g ² t 1 , . . . , t iy1 : s S i . But Lemma 2.4Ža. shows that T¨ L i, 1 is equal to an A-linear combination of terms L j, 1Tw with j F i, w g S i . Thus we have that Lki is a linear combination of terms L1, c1 ??? L i, c i L j, 1Tw with j F i, w g S i . Now L i, c i L j, 1 with 1 F c i F k y 1, j F i is a linear combination of terms L1, d1 ??? L i, d i with 1 F d i F k by Lemma 2.4Žb.. This shows that Lki is a linear combination of terms hL i, d i Tw with 1 F d i F k and h in the subalgebra H ŽWiy1 , u.. Expanding h in the reduced basis of this subalgebra we obtain summands of the form L1, c1 ??? L iy1, c iy 1T¨ L i, c i Tw with ¨ g ² t 1 , . . . , t iy2 :. It remains to apply Lemma 2.4Ža. again to move the T¨ past L i, c i . This completes the induction step and hence the proof of assertion Ža.. For part Žb. we use induction on i. If i s 1 then the result is just part Ža.. So assume that the assertion is proved for all j - i. By induction we have that Lb11 ??? Lbi i is a linear combination of terms L1, c1 ??? L iy1, c iy 1Tw Lbi i with w g ² t 1 , . . . , t iy2 :. By the first part, Lbi i is an A-linear combination of terms L1, c1 ??? L i, c i Tw with c i / 0. Again the result follows by Lemma 2.4. We are now in a position to complete the proof of the proposition. Let 1 / b s Lb11 ??? Lbi i T¨ g B with a i / 0. If i s 0 then b s T¨ is already contained in BX so there is nothing to prove. If i ) 0 then according to Lemma 2.5Žb. we have that b is equal to a linear combination of basis elements L1, c1 ??? L i, c i Tw g BX with c i / 0. In particular, f H Ž b . s 0 by Ž1.6..
3. THE DETERMINANT OF THE GRAM MATRIX OF TYPE GŽ r, 1, n. Let ): H ª H be the A-linear antiautomorphism of H determined by TiU s Ti for 0 F i - n. So, in particular, LUi s L i and TwU s Twy1 for all i with 1 F i F n and all w g S n . Given a set H : H let H U s hU N h g H 4 . In order to show that f H induces a symmetric form on H over A it suffices to exhibit two bases B, C of H over A such that the determinant of the Gram matrix GB C [ Ž² b, c :.B, C s Ž f H Ž bc .. b g B, c g C is a unit in A.
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
281
For the basis B we take the Ariki]Koike basis defined in Ž2.1. above. For the basis C we essentially take BU ; however, for technical reasons Žsee Proposition 3.8 below., the definition is more complicated. Let l 1 s 1 and for i s 2, . . . , n let l i s qy1 Tiy1 l iy1Tiy1. Then l i is the analogue of L i in the subalgebra H Ž S n , q .; indeed, L i s u1 l i if r s 1. We define C [ T¨ l 1d Ž c1 . ??? l nd Ž c n . Lc11 ??? Lcnn N 0 F c i F r y 1, ¨ g S n 4 , where d Ž c . s y1 if c / 0 and d Ž0. s 0. For a given c s Ž c1 , . . . , c n ., the set T¨ l 1d Ž c1 . ??? l nd Ž c n .4¨ g S n is a basis of H Ž S n , q . because each l i is invertible. Consequently, C is a basis of H by Ž2.1.. THEOREM 3.1. Let B and C be as abo¨ e. Then detŽ GB C . s "q l Ž u1 ??? . u r m for some integers l and m. By Proposition 2.2, and the above remarks, the main theorem is a consequence of Theorem 3.1 when W is of type BnŽ r . ; in the next section we deduce the general case from this. The proof of Theorem 3.1 will occupy the whole of this section. We begin with a well-known lemma which follows directly from the relations in H Žsee, for example, w2, Lemma 3.3x.. LEMMA 3.2.
Suppose that 1 F i - n and that 1 F j F n. Then
Ža. L i commutes with L j . Žb. Ti commutes with L j if j / i, i q 1. Žc. Ti commutes with L i L iq1. The elements l 1 , . . . , l n also satisfy similar relations to those given in the lemma; however, we shall not need this. LEMMA 3.3. integers.
Suppose that 1 F i - n and let k and m be non-negati¨ e
Ža. If k G m then myc Lki Lmiy1Tiy1 s Tiy1 Lmi Lkiy1 q Ž q y 1 . Ý kcsmq1 Lci Lkq . iy1
Žb. If k F m then c kq myc . Lki Lmiy1Tiy1 s Tiy1 Lmi Lkiy1 y Ž q y 1 . Ý m cskq1 L i L iy1
Proof. Part Ža. is proved in w6, Lemma 5.10x; the argument is a routine induction on k y m Žuse Lemma 3.2Žc. to start the induction.. To prove Žb., apply the involution ) to Ža. and rearrange.
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MALLE AND MATHAS
Suppose that 1 - i F n and k G 1. Then
COROLLARY 3.4.
Lki s qy1 Tiy1 Lkiy1Tiy1 q qy1 Ž q y 1 .
ky1
Ý Lci Lkyc iy1 Tiy1 . cs1
Proof. When k s 1 this is just the definition of L i . In general, taking m s 0 in part Ža. of the lemma, we see that 1 Lkq s qy1 Lki Tiy1 L iy1Tiy1 i
½
s qy1 Tiy1 Lkiy1 q Ž q y 1 .
k
Ý Lci Lkyc iy1 cs1
y1 s qy1 Tiy1 Lkq1 Ž q y 1. iy1 Tiy1 q q
5
L iy1Tiy1
k
Tiy1 . Ý Lci Lkq1yc iy1 cs1
Recall the trace form f H : H ª A from Ž1.3.. Given h, hX g H we define fH
fH
an equivalence relation s on H by saying that h s hX whenever f H Ž h. s f H Ž hX .. This equivalence relation does not respect multiplication in H ; however, because f H is a trace form, it does have the following crucial property. fH
Ž3.5. Let h, hX g H . Then hhX s hX h. For j s 0, . . . , n let H j be the A-submodule of H with basis
Laj
j
??? La11 Tw N 0 F a i - r , w g S n 4 .
Note that Hn s H and that H j is not an H-module for j / n. However, H j is always a right H Ž S n , q .-module and, by Proposition 2.2, we also have the next result. Ž3.6. Suppose that 1 F j - n, 1 F k F r y 1 and h g Hj . Then fH
H Lkjq1 h s 0.
What we really need to know, however, is how to compute f H Ž Lrjq1 h.. LEMMA 3.7. Suppose that i, j, and k are integers such that 1 F i F j - n and 1 F k F r, and let h g H j . Then fH
Lkjq1 h s q iyjy1 Tj ??? Ti Lki Ti ??? Tj h.
283
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
Proof. For the duration of this proof let Tj??i s Tj ??? Ti and Ti?? j s Ti ??? Tj . By declaring that Tj??iq1 s 1 s Tiq1?? j we can state the lemma when i s j q 1; in which case the lemma asserts that f H Ž Lkjq1 h. s f H Ž Lkjq1 h.. Since this is certainly true, we may now proceed by downwards induction on i F j. Now, by induction and by Corollary 3.4, fH
q jyiq2 Lkjq1 h s qTj??i Lki Ti?? j h ky1
s Tj??i Tiy1 Lkiy1Tiy1Ti?? j h q Ž q y 1 .
Ý Tj??i Lci Lkyc iy1 Tiy1Ti?? j h. cs1 fH
Thus, it suffices to show that Tj??i Lci Lkyc iy1 Tiy1Ti?? j h s 0 whenever 0 - c - k. Fix such an integer c. Then, using the definition of the elements L a together with Lemma 3.2, we find c jyiq2 cy1 y1 y1 Tj??i Lci Lky Tj??i Lci Lky Tj??i L jq1 . h Ž Tiy1 iy1 Tiy1Ti?? j h s q iy1 cy1 c y1 y1 s q jyiq2 Tj??i Ž Lky L i . Tiy1 Tj??i L jq1 h iy1 cy1 1 y1 s q jyiq1 Tj??i Lky Tiy1 L iy1 . Tj??i L jq1 h Ž Lcy iy1 i 1 ky cy1 y1 s q jyiq1 Tj??i Lcy L iy1 Tiy1Tj??i L jq1 L iy1 h. i
Now Tay1 s qy1 ŽTa q 1 y q . for 1 F a - n. Therefore, by Lemma 3.3, together with Lemma 3.2Ža., there exist scalars a L g A such that 1 ky cy1 y1 Lcy L iy1 Tiy1Tj??i s i
Ý
jq 1 ??? L b 1 gB U LsTw Lbjq1 1
jq 1 ??? Lb 1 , a L Tw Lbjq1 1
where bjq1 F max c y 1, k y c y 14 F k y 2 whenever a L / 0. Consequently, c Tj??i Lci Lky iy1 Tiy1Ti?? j h s
jq 1 ??? Lb 1 L q jyiq1a L Tj??i Tw Lbjq1 1 jq1 L iy1 h
Ý jq 1 ??? L b1 LsTw Lbjq1 1
fH
s
jq 1 ??? Lb 1 L q jyiq1 a L L1qb jq1 1 iy1 hTj??i Tw
Ý jq 1 LsTw Lbjq1
???
Lb11 fH
by Ž3.5.. However, 1 q bjq1 F k y 1 - r; so Tj??i Lci Lkyc iy1 Tiy1Ti?? j h s 0 by Ž3.6. as required.
284
MALLE AND MATHAS
Recall the element l jq1 s qyj Tj ??? T1T1 ??? Tj . Let j be an integer such that 1 F j - n and suppose
PROPOSITION 3.8. that h g H j . Then
fH
r
Lrjq1 h s Ž y1 . u1 ??? u r h l jq1 . Proof. By Lemma 3.7, f H Ž Lrjq1 h. s qyj f H ŽTj ??? T1T0r T1 ??? Tj h.. Therefore, since ŽT0 y u1 . ??? ŽT0 y u r . s 0, we may write f H Ž Lrjq1 h. as an A-linear combination of terms f H ŽTj ??? T1T0k T1 ??? Tj h. for k s 0, . . . , r y 1. However, by Lemma 3.7 and Ž3.6., if 0 - k - r then fH
fH
Tj ??? T1T0k T1 ??? Tj h s q j Lkjq1 h s 0. fH
Therefore, Lrjq1 h s Žy1. r qyj u1 ??? u r hTj ??? T1T1 ??? Tj s Žy1. r u1 ??? u r h l jq1 as required. We can now prove the main result of this section. Proof of Theorem 3.1. Recall the bases B and C of H and the Gram matrix GB C s Ž f H Ž bc .. b g B, c g C from the beginning of this section. An arbitrary element of B is of the form Lbnn ??? Lb11 Tw for some w g S n and some integers 0 F bi - r. In the matrix GB C order the elements of B lexicographically according to the exponents Ž bn , . . . , b1 ., and similarly for the elements of C. For j s 0, . . . , n let Bj s B l Hj and C j s C l HjU and define Gj to be the Gram matrix GB j C j s Ž f H Ž bc .. b g B j, c g C j . Then GB C s Gn and our ordering of the elements of B and C is such that the matrix Gj sits in the top left hand corner of the matrix Gjq1 for each 0 F j - n. Suppose that 0 F j - n and let a s Žy1. r u1 ??? u r . Then we claim that, as an r = r-block matrix, Gjq1 has the form
Gj
0
???
0
0
0 .. . 0
0 .. . 0
??? ? ?? ? ??
0 ? ?? )
a Gj
0
a Gj
)
???
) .. . )
0
.
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
285
d Ž c1 . d Ž c jq 1 . c 1 jq 1 ??? Lb 1 T jq 1 To see this let b s Lbjq1 ??? l jq1 L1 ??? Lcjq1 1 w and c s T¨ l 1 be arbitrary elements of Bjq1 and C jq1 , respectively. Then jq 1 ??? Lb 1 T T l d Ž c 1 . ??? l d Ž c jq 1 . Lc 1 ??? Lc jq 1 bc s Lbjq1 1 w ¨ 1 1 jq1 jq1
fH
jq 1 qc jq 1 Lb jqc j ??? Lb 1 qc 1 T T l d Ž c 1 . ??? l d Ž c j . l d Ž c jq 1 . s Lbjq1 j 1 w ¨ 1 j jq1
by Ž3.5. and Lemma 3.2Ža.. If bjq1 s c jq1 s 0 the assertion is just the definition of Gj Žnote that d Ž0. s 0.. If 0 - bjq1 q c jq1 - r then f H Ž bc . s 0 by Ž3.6.. Finally, if bjq1 q c jq1 s r then by Proposition 3.8 fH
r
d Ž c jq 1 . bc s Ž y1 . u1 ??? u r Lbj jqc j ??? Lb11qc1 Tw T¨ l 1d Ž c1 . ??? l jd Ž c j . l jq1 l jq1 r
s Ž y1 . u1 ??? u r Lbj jqc j ??? Lb11qc1 Tw T¨ l 1d Ž c1 . ??? l jd Ž c j . , since d Ž c jq1 . s y1 because c jq1 ) 0. This proves the claim. Therefore, det Gjq1 s "Ž u1 ??? u r . ry1 Ždet Gj . r for all 0 F j - n. Consequently, det GB C s det Gn s "Ž u1 ??? u r . m Ždet G0 . k for some integers k and m. However, G0 s Ž f H ŽTw T¨ .. ¨ , w g S n and it is well known, and easy to prove, that f H ŽTw T¨ . s q ldw ¨ y1 , where l is the length of a reduced expression for w. Hence, det G0 s "q z for some integer z. This completes the proof of Theorem 3.1.
4. THE DETERMINANT OF THE GRAM MATRIX OF TYPE GŽ r, 2, 2. For any divisor p of r let Wn, p be the normal subgroup of index p of Wn generated by the n q 1 reflections
t 0p , ty1 0 t 1 t 0 , t 1 , . . . , t ny1 4 if p - r, respectively, the n reflections
ty1 0 t 1 t 0 , t 1 , . . . , t ny1 4 if p s r. Thus Wn, p is the imprimitive complex reflection group GŽ r, p, n. in the notation of Shephard and Todd. We would like to introduce a symmetrizing form on the generic cyclotomic algebra H ŽWn, p , u. defined in w5x. This coincides with the algebra introduced by Ariki in w1x except for the case where n s 2, and both r and p are even, where the generic algebra has one additional parameter. In this latter case, no quasi-symmetric basis had been given in w3x, so we need to treat this algebra separately.
286
MALLE AND MATHAS
So, suppose that r is even, n s p s 2, and set d [ rr2. Then W2, 2 has a presentation described by the diagram
"
D Ž2d. :
Ž 4.1.
#
"
d "
Let H s H ŽW2, 2 , u., where u s Ž u1 , . . . , u d ; q; qX ., be the corresponding cyclotomic algebra, generated as associative unital Zwu, uy1 x-algebra by elements R, S, T subject to the relations
Ž T y u1 . ??? Ž T y u d . s 0 s Ž R y q . Ž R q 1 . s Ž S y qX . Ž S q 1 . , RST s STR s TRS Žsee w4, 4Ax.. We let U [ RS and V [ Ž qX y 1.UrqX q q y 1. Note that T and U commute and the element TU is central in H . The following result is inspired by w1, Lemma 1.2x. Let j, k be non-negati¨ e integers. Then
LEMMA 4.2.
Ž a. ST j s Ž qqX .
yj
T jU 2 j S y
1 q
jy1
T jUV
Ý Ž qqX . yiU 2 i ,
is0
Ž b. SU k s Ž qqX . Uyk S q k
1 q
ky1
UV
Ý Ž qqX . kyiU 2 iyk ,
is0
Ž c.
¡y 1 T V j
X
ST U s Ž qq . j
k
ky j
j
T U
2 jyk
S q~
q
1
¢q T V j
jyky1
Ý Ž qqX . yi U 2 iq1qk
is0 y1
Ý Ž qqX . yi U 2 iq1qk
if j G k, if j - k.
isjyk
Proof. We prove Ža. by induction on j. The result trivially holds if j s 0. If j s 1 then ST s STRRy1 s s Ž qqX .
y1
1 q
TUR y
TU 2 S y
1 q
qy1 q
TUV ,
TU s
1 q
TU 2 Sy1 y
qy1 q
TU
287
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
as claimed. The induction hypothesis and the case j s 1 now yield ST jq1 s Ž qqX . s Ž qqX .
yj
T jU 2 j ST y
yjy1
1 q
jy1
T jq1U
Ý Ž qqX . yiU 2 i V
is0
T jq1 U 2 jq2 S y
1 q
j
T jq1 U Ý Ž qqX .
yi
U 2iV ,
is0
proving the first part. In part Žb. the case k s 0 is again obvious. If k s 1 then SU s SRS s Ž qX Sy1 q qX y 1 .Ž qRy1 q q y 1 . S s qqX Uy1 S q Ž q y 1 . qX q Ž qX y 1 . Ž qRy1 q q y 1 . S s qqX Uy1 S q Ž q y 1 . qX q Ž qX y 1 . U s qqX Uy1 S q qX V . The inductive step is verified by the same type of calculation as in the first part. The last part is an immediate consequence of parts Ža. and Žb.. Note that T, U generate a commutative subalgebra of H . Since it is known that H is Zwu, uy1 x-free of rank < W2, 2 < by w4, Satz 4.7x we conclude as there that
T jU k , T jU k S N 0 F 2 j y k , k F r y 1 4 is a Zwu, uy1 x-basis of H Žcompare also with w1, Proposition 1.4x where the case q s qX is considered.. We proceed to show that this basis has good properties. PROPOSITION 4.3. The basis C s T jU k , T jU k S N 0 F 2 j y k, k F r y 1 4
Ž 4.4.
of H s H ŽW2, 2 , u. is quasi-symmetric. Proof. We have to show that the linear form f H : H ª Zwu, uy1 x defined by f H Ž1. s 1, f H Ž b . s 0 if 1 / b g C is a trace form. Since it specializes to the standard symmetrizing form of the complex reflection group W2, 2 the induced symmetric form on H is then automatically non-degenerate over the quotient field of A. We will have to consider elements of H of the form T jU k S l with 0 F k F r y 1, l g 0, 14 , and j an arbitrary integer. First note that on such elements fH Ž T jU k S l . s 0
if k / 0 or l / 0.
Ž 4.5.
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Indeed, if T jU k S l f C then by the relation on T we may replace T j by a linear combination of elements T jqi or T jyi, with 1 F i F d. Induction and the definition of f H now show the claim. Since R and S are invertible in H it clearly suffices to prove that f H Ž bL. s f H Ž Lb . for b g C and L g S, U, TU 4 . This is immediate for the central element L s TU. Now assume L s S. Inspection shows that if 0 F 2 j y k, k F r y 1 then all monomials on the right hand side of Lemma 4.2Žc. lie in C. Thus, Lemma 4.2Žc. gives the decomposition in C of ST jU k . By the definition of f H it follows that f H Ž S ? T jU k . s 0 s f H ŽT jU k ? S .. Similarly, f H ŽT jU k S ? S . equals fH Ž S ? T jU k S . s
½
qX 0
if j s k s 0, else.
So finally assume that L s U. As U commutes with T, the assertion is again clear if b s T jU k . For b s T jU k S we have to compare the value of f H on T jU kq 1S and on T jU k SU s qqX T jU ky 1S q qX Ž q y 1 . T jU k q Ž qX y 1 . T jU kq 1 Ž 4.6. Žby Lemma 4.2Žb... If k / 0, r y 1, and 2 j y k / 0, r y 1 then all monomials in the above expressions lie in C and f H vanishes on both sides. If 2 j y k g 0, r y 14 then f H again vanishes by Ž4.5.. We consider the two remaining cases separately. First assume that k s 0. We have f H ŽU ? T j S . s f H Ž ST jU . by what we proved for S, so the result is true if j s 0. For j ) 0 ST jU s TUST jy1 s Ž qqX .
1y j
T jU 2 jy1 S y
1 q
jy2
Ý Ž qqX . yiU 2 i
T jU 2 V
is0
by Lemma 4.2Ža.. All monomials appearing in the right-hand side are non-trivial basis elements, so f H vanishes. On the other hand we have the equality dy1
SU s S
Ý
dy1
a l T lq1U s
ls0 dy1
s
Ý ls0
Ý al TUST l ls0
a l Ž qqX .
ž
yl
T lq1 U 2 lq1 S y
1 q
ly1
T lq1 U 2 V Ý Ž qqX . is0
yi
U 2i
/
Žwith suitable constants a l . since TU is central and by Lemma 4.2Ža.. Here, f H is zero by Ž4.5..
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
289
It remains to consider k s r y 1. Lemma 4.2Ža. gives ST d s Ž qqX . y
yd
T d U r Ž S y Ž qX y 1 . .
qy1 q
T d U ry1 Ž qqX .
1y d
y
1 q
dy2
T d UV
Ý Ž qqX . yiU 2 i . Ž 4.7.
is0
The last two terms on the right-hand side are of the type Ž4.5.. On the other hand, replacing T d by a linear combination of T i with 0 F i F d y 1 and again applying Lemma 4.2Ža. shows that the left-hand side is a sum of elements of the form considered in Ž4.5.. Multiplication of both sides by TyŽ dyj. does not change this fact, and thus f H ŽT jU r S . s Ž qX y 1. f H ŽT jU r .. But now Ž4.6., with k s r y 1, gives f H Ž T jU ry1S ? U . s Ž qX y 1 . f H Ž T jU r . s f H Ž T jU r S . s f H Ž U ? T jU ry1S . as required. We write GC for the Gram matrix of f H with respect to the pair of bases C, CU , where CU is obtained from C by reversing the order of the factors in each element. PROPOSITION 4.8. Let C be the basis of H defined in Ž4.4.. Then detŽ GC . is in¨ ertible in A s Zwu, uy1 x. In particular H is symmetric o¨ er A. Proof. Elements of C are of the form T jU k S l for some j, k, l with 0 F k, 2 j y k F r y 1 and l g 0, 14 . We order C lexicographically with respect to Ž k, l, j .. Then Ž4.5. shows that the Gram matrix has the shape
G1 0 .. . 0
0 0 .. . 0
??? ??? ? ?? ? ??
0 0 ? ?? )
0
Gr
)
???
0 G2 ) .. . )
0
Žas an r = r-block matrix.. Again by Ž4.5. we see that G1 is a block diagonal 2 = 2-matrix, and the blocks on the diagonal are the same as for the algebra H ŽW1 , u.. Thus the determinant of G1 has the required form. The matrices Gi , 2 F i F r, can also be considered as 2 = 2-block matri-
290
MALLE AND MATHAS
ces, with entries f H ŽT j1qj 2 U r S l 1ql 2 . where l 1 , l 2 g 0, 14 . Application of Ž4.7. shows that Gi has the block form
ž
GiX
Ž qX y 1 . GiX
Ž qX y 1 . GiX
Ž qX q Ž qX y 1. 2 . GiX
/
.
Here GiX is the matrix with entries f H ŽT j1qj 2 U r .. Subtraction of Ž qX y 1. times the first block of r columns from the second block of r columns thus yields det Gi s qX detŽ GiX . 2 . To evaluate the determinant of GiX we need a further calculation. Application of Ž4.7. gives 1 T l U r s X T l U r Ž S y Ž qX y 1 . . S q s
1 q
X
d Ž qqX . T lyd ST d q
ž
qy1 q
T d U ry1 Ž qqX .
1yd
dy2 1 yi y T d UV Ý Ž qqX . U 2 i S q is0
/
s q Ž qqX .
dy 1
T lydST dS q Ž q y 1 . T l U ry1 S
dy2
y T l UV
Ý Ž qqX . yiy1U 2 iS.
is0
According to Ž4.5. the form f H vanishes on all summands on the right-hand 1 j side, except possibly the first. Hence, expanding T d s Ý dy js0 a j T , we have fH
T l U r s q Ž qqX .
dy 1
dy1
T lydS
Ý a jT jS js0
s q Ž qqX .
dy 1
dy1
T lyd
ž
js0 fH
s q Ž qqX .
dy 1
1
jy1
Ý a j Ž qqX . yj T jU 2 jS y q T jUV Ý Ž qqX . yiU 2 i fH
is0
/
S
T lyda0 S 2 s Ž qqX . u1 ??? u d T lyd d
by Lemma 4.2Ža. and Ž4.5.. Note that this is the analogue of Proposition 3.8 in this case. In particular this last equation shows that the determinant of GiX is the same as the one occurring in H ŽW1 , u. Žup to units in A.. The assertion follows. Let f H : H ª A be the symmetrizing form induced by the quasi-symmetric basis C. Over the quotient field K [ QuotŽ A. we may decompose f H as a sum fH s Ý dx x x gIrr Ž HK .
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
291
of irreducible characters of HK [ H m K, with dx g K. The coefficients dx are called the relati¨ e degrees associated to f H , respectively, to the quasisymmetric basis C. Clearly we have
Ý
x gIrr Ž HK .
dx x Ž c . s 0
for all elements c / 1 of a quasi-symmetric basis with respect to f H , and Ý dx x Ž1. s 1. Using this property of the relative degrees the first author in w8x computed candidates for the relative degrees attached to a nice symmetrizing form on H . We can now verify: COROLLARY 4.9. The relati¨ e degrees of the symmetric form induced by the quasi-symmetric basis C are those gi¨ en in w8, Theorem 3.9x. Proof. According to the result in w8x it suffices to show that f H vanishes on the elements
RT jU j , ST jU j N 0 F j F d y 1 4 j T iqjU j N 0 - 2 i q j F r y 1 4 of H . But RT jU j s T jU jq1 Sy1 s Ž1rqX .T jU jq1 Ž S y Ž qX y 1.., on which f H vanishes by Ž4.7., ST jU j s T jU j S g C _ 14 , and also T iqjU j g C, so the result follows.
5. THE CYCLOTOMIC ALGEBRAS H Ž GŽ r, p, n., u. Finally, we consider the general case where Wn, p is an arbitrary imprimitive complex reflection group GŽ r, p, n.. Let H ŽWn, p , u. be the generic cyclotomic algebra associated to Wn, p in w5x. This coincides with the algebra defined by Ariki w1, Proposition 1.6x except in the case where n s 2 and r, p are both even. THEOREM 5.1. Let W s GŽ r, p, n. be an imprimiti¨ e complex reflection group. Then the associated generic cyclotomic algebra H s H ŽW, u. carries a canonical symmetrizing form f H o¨ er Zwu, uy1 x. This form ¨ anishes on all non-identity elements of the Ariki]Koike basis of H , while f H Ž1. s 1. In the case p s 1 it also ¨ anishes on all non-identity elements of any reduced basis of H . Proof. We reduce this to the case p s 1 Žrespectively p s 2 if n s 2 and r, p are even. where such a symmetrizing form was found in Theorem 3.1 Žresp. Proposition 4.8.. Assume first that either n ) 2 or one of r or p is odd. Let d [ rrp, z p [ expŽ2p irp. and A˜ [ Zw z p , u, uy1 x. Then according to Ariki w1, Proposition 1.6x the algebra H m A˜ can be identified
292
MALLE AND MATHAS
˜ with a subalgebra of the specialized A-algebra H˜s H ŽWn , ˜ u., with parameters ˜u s Ž u1 , . . . , u d , z p u1 , . . . , z p u d , . . . , z ppy 1 u1 , . . . , z ppy 1 u d ; q . , Ž 5.2. generated by
T0p , Ty1 0 T1T0 , T1 , . . . , Tny1 4 . Let f˜ be the linear form on H˜ obtained by specializing the symmetrizing form constructed in Section 3. Since the specialization to the group algebra factorizes through the specialization Ž5.2. the form f˜ gives rise to a non-degenerate symmetric form on H˜. In particular, all irreducible charac˜ This implies that all ters of H˜m QŽu. occur with non-zero multiplicity in f. irreducible characters of the subalgebra H m A˜ of H˜ occur with non-zero ˜ Thus, fH is a symmetrizing multiplicity in the restriction f H of f˜ to H m A. ˜ form for H over the quotient field of A. It is shown in w1, Proposition 1.6x that
Lan
n
??? La11 Tw N 0 F a i F r y 1,
Ý ai ' 0 Ž mod p . , w g S n 4
˜ this gives a basis of H m A˜ which is is an A-basis of H . Tensoring with A, a subset of the Ariki]Koike basis of the cyclotomic algebra H˜ ; it follows ˜ However, the Gram determithat f H is a symmetrizing form for H m A. ˜ nants for the bases of H and H m A are equal and are elements of A, so f H is a symmetrizing form for H Žover A., as required. In the case where n s 2 and r and p are even, we embed H instead into a suitable specialization H ŽW2, 2 , ˜ u. of the generic algebra of W2, 2 studied Ž w in Section 4 see 8, Proposition 3.8x.. The remainder of the argument is then the same as in the general case. Remark 5.3. We expect that f H should also vanish on non-identity reduced basis elements in the general case p / 1. In this context note that a word which is reduced in Wn, p does not necessarily have to be reduced in Wn . Remark 5.4. In w7, Verm. 2.20 and Satz 5.13x the first author put forward a conjecture on the values of the relative degrees dx associated to a nice symmetrizing form on H ŽWn, p .. Using the calculations in w8, Theorem 3.4x and Corollary 4.9 it follows that this conjecture is correct if n s 2 for the symmetrizing form studied in this paper. We expect that in general the relative degrees attached to the symmetrizing form f H above are those predicted in w7x.
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
293
Finally, suppose that O is an integral domain and let HO ŽWn, p , ˆ u. be the specialized cyclotomic Hecke algebra of Wn, p with parameters ˆ u g O. Given a set of indeterminates u over Z we can consider O as a Zwux-module by specifying that u acts as ˆ u. Then HO ŽWn, p , ˆ u. ( HZwuxŽWn, p , u. mZwux O . Therefore, by Theorem 5.1, we have the following. COROLLARY 5.5. Let HO ŽWn, p , ˆ u. be a specialized cyclotomic Hecke algebra o¨ er O . Then HO ŽWn, p , ˆ u. is a symmetric algebra with respect to f H if and only if the parameters ˆ u are in¨ ertible in O . In w6x an analogue of the q-Schur algebra for a cyclotomic Hecke alˆ gebra HO of type BnŽ r . was introduced. Key to the main results of that paper is the calculation of the ‘‘double annihilator’’ of certain elements of H Žsee w6, Theorem 5.16x.. When the parameters u are invertible elements of O , the required result of w6x is an immediate consequence of Corollary 5.5 Žsignificantly simplifying the argument there.. Interestingly, the results of w6x require only that qˆ be invertible in O . This suggests that HO may be a quasi-Frobenius algebra so long as qˆ is an invertible element of O .
REFERENCES 1. S. Ariki, Representation theory of a Hecke algebra of GŽ r, p, n., J. Algebra 177 Ž1995., 164]185. 2. S. Ariki and K. Koike, A Hecke algebra of ŽZrrZ. X Sn and construction of its irreducible representations, Ad¨ . in Math. 106 Ž1994., 216]243. 3. K. Bremke and G. Malle, Reduced words and a length function for GŽ e, 1, n., Indag. Math. 8 Ž1997., 453]469. 4. M. Broue 212 Ž1993., 119]189. ´ and G. Malle, Zyklotomische Heckealgebren, Asterisque ´ 5. M. Broue, ´ G. Malle, and R. Rouquier, On complex reflection groups and their associated braid groups, in ‘‘Representations of Groups’’ ŽB. N. Allison and G. H. Cliff, Eds.., CMS Conference Proceedings, Vol. 16, pp. 1]13, Amer. Math. Soc., Providence, 1995. 6. R. Dipper, G. James, and A. Mathas, Cyclotomic q-Schur algebras, Math. Z., in press. 7. G. Malle, Unipotente Grade imprimitiver komplexer Spiegelungsgruppen, J. Algebra 177 Ž1995., 768]826. 8. G. Malle, Degres cyclotomiques associees ´ relatifs des algebres ` ´ aux groupes de reflexions ´ complexes de dimension deux, in ‘‘Finite Reductive Groups: Related Structures and Representations’’ ŽM. Cabanes, Ed.., Progr. Math, Vol. 141, pp. 311]332, Birkhauser, ¨ Boston, 1997.
JOURNAL OF ALGEBRA ARTICLE NO.
Symmetric Cyclotomic Hecke AlgebrasU Gunter Malle† IWR, Im Neuenheimer Feld 368, Uni¨ ersitat ¨ Heidelberg, D-69120, Heidelberg, Germany
and Andrew Mathas ‡ School of Mathematics F07, Uni¨ ersity of Sydney, Sydney, NSW 2006, Australia Communicated by Michel Broue´ Received September 30, 1997
In this paper we prove that the generic cyclotomic Hecke algebras for imprimitive complex reflection groups are symmetric over any ring containing inverses of the parameters. For this we show that the determinant of the Gram matrix of a certain canonical symmetrizing form introduced by K. Bremke and G. Malle Ž Indag. Math. 8 Ž1997., 453]469. is a unit in any such ring. On the way we show that the Ariki]Koike bases of these algebras are also quasi-symmetric. Q 1998 Academic Press
0. INTRODUCTION AND STATEMENT OF THE MAIN RESULT Work of Broue, ´ Michel, and the first author on l-blocks in finite groups of Lie type suggests that the decomposition of certain Lusztig induced characters should be governed by suitable complex reflection groups. This led Broue ´ and the first author in w4x to introduce the concept of the cyclotomic Hecke algebra H ŽW, u. associated to a complex reflection U
The first author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft. The second author thanks the Universitat ¨ Heidelberg for its hospitality during the writing of this paper and also the Sonderforschungsbereich 343 at the Universitat ¨ Bielefeld for financial support. † E-mail: [email protected]. ‡ E-mail: [email protected]. 275 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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MALLE AND MATHAS
group W. Conjecturally specializations of this algebra should occur as endomorphism algebras of Lusztig induced characters, thus explaining the aforementioned observation Žsee w4x.. This general conjecture has been proved only in a small number of cases. Nevertheless, several structural and numerical consequences of this conjecture have been verified. One of the properties of cyclotomic algebras implied by the general conjecture is that H should carry a canonical symmetrizing form making it into a symmetric algebra over the ground ring A in which just the parameters are inverted. Such a canonical symmetrizing form over the field of fractions of A was constructed in w3x for all but finitely many of the cyclotomic algebras associated to irreducible complex reflection groups. In this paper we prove that pairs of dual bases with respect to this form are defined over A: THEOREM. The generic cyclotomic algebra H s H ŽW, u. of an irreducible imprimiti¨ e complex reflection group W is symmetric o¨ er Zwu, uy1 x. The proof of this result will be given in the subsequent sections. There we will also give more precise statements about properties of the corresponding symmetrizing form. The assertion of the main theorem is well known in the case of Iwahori]Hecke algebras, i.e., for cyclotomic algebras for real reflection groups. In that case a dual basis can immediately be written in terms of the standard basis of H and the assertion follows easily. It turns out that the situation for the complex case is more subtle. We would like to point out that it is still open whether the cyclotomic algebras for all of the finitely many primitive complex reflection groups enjoy the properties stated in the theorem.
1. THE SYMMETRIZING FORM ON H ŽWn , u. In this section we recall the definition of the cyclotomic Hecke algebra H ŽWn , u. and the construction of a symmetrizing form on H ŽWn , u. m QŽu. from w3x. Let V be the n-dimensional unitary space V s ² e1 , . . . , e n : with the standard unitary inner product. Fix r ) 1, and let Wn s GŽ r, 1, n. ; GLŽ V . be the imprimitive complex reflection group on V generated by the reflections t 1 , . . . , t ny1 of order 2 with roots e2 y e1 , . . . , e n y e ny1 , respectively, and a Žcomplex. reflection t 0 of order r with root e1. On these
277
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
generators, Wn has a presentation given by the diagram BnŽ r . :
r " " t t 0
1
???
" t 2
Ž 1.1.
".
t ny1
The elements t 1 , . . . , t ny1 generate a subgroup of Wn isomorphic to the symmetric group S n , and form a set of standard generators of this Coxeter group. We will consider S n as a subgroup of Wn via this embedding. Let u s Ž u1 , . . . , u r , q . be indeterminates and A [ Zwu, uy1 x, where y1 y1 . we write uy1 [ Ž uy1 . The generic cyclotomic algebra H [ 1 , . . . , ur , q Ž . H Wn , u attached to Wn is the free unital associative A-algebra on generators T0 , T1 , . . . , Tny1 , subject to the braid relations implied by the diagram Ž1.1. and the deformed order relations
Ž T0 y u1 . ??? Ž T0 y u r . s 0 s Ž Ti y q . Ž Ti q 1 . ,
1FiFny1
Žsee w4, 4.1x.. Under the specialization u k ¬ zr k
Ž1 F k F r . ,
q ¬ 1,
Ž 1.2.
where zr [ expŽ2p irr ., the cyclotomic algebra specializes to the group algebra of Wn over Zw zr x. Further, it is known that H is a free A-algebra of rank < Wn < Žsee w2x.. Thus the subalgebra generated by T1 , . . . , Tny1 can and will be identified with the generic Iwahori]Hecke algebra H Ž S n , q . of type A ny 1. For x s Ž x 1 , . . . , x l ., x i g t 0 , t 1 , . . . , t ny14 , a reduced expression of an element x s x 1 ??? x l in Wn , let Tx [ Tx 1 ??? Tx l . Note that it is not in general true that two reduced expressions x, x X of the same element x g Wn give rise to identical elements Tx , Tx X in H . Nevertheless, this property does hold if x g S n F Wn . Let X s x N x g W 4 be an arbitrary set of reduced expressions for the elements of W. Then it is known that the set BX [ Tx N x g X 4 is an A-basis of H Žsee, for example, w3, Proposition 2.4x.. Any such basis will be called a reduced basis of H . Choose one such reduced basis BX . Let f H : H ª A be the linear form defined on BX by f H : H ª A,
f H Ž Tx . s
½
1 0
if x s 1, otherwise.
Ž 1.3.
It was shown in w3, 2Bx that the linear form f H does not depend on the particular reduced expressions chosen in the definition of the reduced basis BX . Hence f H is defined unambiguously on H . Moreover, it was
278
MALLE AND MATHAS
proved in w3, Proposition 2.7x that f H is a trace form. Thus the bilinear form ² , : : H H ª A,
Ž h1 , h 2 . ¬ f H Ž h1 h 2 .
Ž 1.4.
is symmetric. Clearly the specialization of ² , : to the group algebra of Wn is the standard symmetrizing form which sends Ž w 1 , w 2 . to the coefficient of 1 in the product w 1w 2 . Since the algebra HK [ H ŽWn , u. mA K over K s QuotŽ A. is split semisimple w2x, this implies: THEOREM 1.5 w3, Theorem 2.8x. The linear form f H defines a non-degenerate symmetric form ² h1 , h 2 : [ f H Ž h1 h 2 .
for h1 , h 2 g HK
on HK . This satisfies f H Ž Tx . s 0
for all reduced expressions x of non-identity elements of Wn .
Ž 1.6. We will show that f H actually defines a non-degenerate symmetric form on H over A. This seems to be important for a further study of this algebra. Let O be a domain and H an unital associative O-algebra. Recall from w4, 1Cx that an O-basis B of H with 1 g B is called quasi-symmetric if the linear form f : H ª O defined by setting f Ž b . s d 1 b for b g B is a trace form and via Ž1.4. gives rise to a non-degenerate symmetric form on H . We then also say that B is quasi-symmetric with respect to f. In this terminology, the above theorem can be rephrased to saying that any reduced basis of H ŽWn , u.K is quasi-symmetric with respect to the form f H defined above.
2. THE ARIKI]KOIKE BASIS OF H IS QUASI-SYMMETRIC It will be convenient to introduce a further quasi-symmetric basis of H s H ŽWn , u.. For this let L1 [ T0 and inductively L i [ qy1 Tiy1 L iy1Tiy1 for 2 F i F n. It was shown by Ariki and Koike w2, Theorem 3.10x that B [ Lb11 ??? Lbnn Tw N 0 F bi F r y 1, w g S n 4
Ž 2.1.
is a basis of H ŽWn , u.. Here Tw denotes the standard basis element in the Iwahori]Hecke subalgebra H Ž S n , q . of H ŽWn , u.. Note that B is nonreduced if r ) 1 and n ) 1. We can now prove the analogue of Ž1.6. for this basis.
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
279
PROPOSITION 2.2. The Ariki]Koike basis B is quasi-symmetric with respect to f H , i.e., it satisfies fH Ž b . s 0
for all 1 / b g B.
Ž 2.3.
Proof. We compare B to a particular reduced basis BX of H and show that any 1 / b g B is a linear combination of non-identity basis elements from BX . The assertion then follows from Theorem 1.5. To describe BX we introduce the elements Lk , a [
½
Tky 1 ??? T1T0a 1
for a ) 0, for a s 0,
for 1 F k F n, 0 F a F r y 1. According to w3, Lemma 1.5x the set BX [ L1, a1 ??? L n , a nTw N 0 F a i F r y 1, w g S n 4 is a reduced basis of H Žthe corresponding set X of reduced words is clear.. Furthermore, the following relations are proved in w3, Lemma 2.3x: LEMMA 2.4. We ha¨ e
Ž a.
¡L
Ti L k , a s
~L
if i ) k or a s 0, if i s k , a / 0,
k , aTi kq 1, a
qL ky 1, a q Ž q y 1 . L k , a k , aTiq1
¢L
if i s k y 1, a / 0, if i - k y 1, a / 0,
for 1 F i F n y 1, 1 F k F n.
Ž b. L k , a L kym , c s L kymy1, c L k , aT1 c
q Ž q y 1.
Ý Ž Lky my1, aqcyi Lk , i y Lkymy1, i Lk , aqcyi . is1
for a, c / 0, k y m ) 1. LEMMA 2.5.
Let i G 1.
Ža. Let 1 F k F r y 1. Then Lki is equal to an A-linear combination of basis elements L1, c1 ??? L i, c i Tw g BX with 0 F c j F r y 1, 1 F c i F k, w g ² t 1 , . . . , t iy1 :.
280
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Žb. Let 1 F bi F r y 1. Then Lb11 ??? Lbi i is equal to an A-linear combination of basis elements L1, c1 ??? L i, c i Tw g BX with 0 F c j F r y 1, c i / 0, w g ² t 1 , . . . , t iy1 :. Proof. According to our definitions we have Lki s Ž q 1y i L i, 1T1 ??? Tiy1 . k . Thus assertion Ža. is certainly correct if k s 1. Now assume that Ža. has been proved for all exponents strictly smaller than k. Hence Lki s 1 Lky L i is a linear combination of terms L1, c1 ??? L i, c i T¨ L i, 1Tw with 1 F c i i F k y 1 and ¨ , w g ² t 1 , . . . , t iy1 : s S i . But Lemma 2.4Ža. shows that T¨ L i, 1 is equal to an A-linear combination of terms L j, 1Tw with j F i, w g S i . Thus we have that Lki is a linear combination of terms L1, c1 ??? L i, c i L j, 1Tw with j F i, w g S i . Now L i, c i L j, 1 with 1 F c i F k y 1, j F i is a linear combination of terms L1, d1 ??? L i, d i with 1 F d i F k by Lemma 2.4Žb.. This shows that Lki is a linear combination of terms hL i, d i Tw with 1 F d i F k and h in the subalgebra H ŽWiy1 , u.. Expanding h in the reduced basis of this subalgebra we obtain summands of the form L1, c1 ??? L iy1, c iy 1T¨ L i, c i Tw with ¨ g ² t 1 , . . . , t iy2 :. It remains to apply Lemma 2.4Ža. again to move the T¨ past L i, c i . This completes the induction step and hence the proof of assertion Ža.. For part Žb. we use induction on i. If i s 1 then the result is just part Ža.. So assume that the assertion is proved for all j - i. By induction we have that Lb11 ??? Lbi i is a linear combination of terms L1, c1 ??? L iy1, c iy 1Tw Lbi i with w g ² t 1 , . . . , t iy2 :. By the first part, Lbi i is an A-linear combination of terms L1, c1 ??? L i, c i Tw with c i / 0. Again the result follows by Lemma 2.4. We are now in a position to complete the proof of the proposition. Let 1 / b s Lb11 ??? Lbi i T¨ g B with a i / 0. If i s 0 then b s T¨ is already contained in BX so there is nothing to prove. If i ) 0 then according to Lemma 2.5Žb. we have that b is equal to a linear combination of basis elements L1, c1 ??? L i, c i Tw g BX with c i / 0. In particular, f H Ž b . s 0 by Ž1.6..
3. THE DETERMINANT OF THE GRAM MATRIX OF TYPE GŽ r, 1, n. Let ): H ª H be the A-linear antiautomorphism of H determined by TiU s Ti for 0 F i - n. So, in particular, LUi s L i and TwU s Twy1 for all i with 1 F i F n and all w g S n . Given a set H : H let H U s hU N h g H 4 . In order to show that f H induces a symmetric form on H over A it suffices to exhibit two bases B, C of H over A such that the determinant of the Gram matrix GB C [ Ž² b, c :.B, C s Ž f H Ž bc .. b g B, c g C is a unit in A.
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
281
For the basis B we take the Ariki]Koike basis defined in Ž2.1. above. For the basis C we essentially take BU ; however, for technical reasons Žsee Proposition 3.8 below., the definition is more complicated. Let l 1 s 1 and for i s 2, . . . , n let l i s qy1 Tiy1 l iy1Tiy1. Then l i is the analogue of L i in the subalgebra H Ž S n , q .; indeed, L i s u1 l i if r s 1. We define C [ T¨ l 1d Ž c1 . ??? l nd Ž c n . Lc11 ??? Lcnn N 0 F c i F r y 1, ¨ g S n 4 , where d Ž c . s y1 if c / 0 and d Ž0. s 0. For a given c s Ž c1 , . . . , c n ., the set T¨ l 1d Ž c1 . ??? l nd Ž c n .4¨ g S n is a basis of H Ž S n , q . because each l i is invertible. Consequently, C is a basis of H by Ž2.1.. THEOREM 3.1. Let B and C be as abo¨ e. Then detŽ GB C . s "q l Ž u1 ??? . u r m for some integers l and m. By Proposition 2.2, and the above remarks, the main theorem is a consequence of Theorem 3.1 when W is of type BnŽ r . ; in the next section we deduce the general case from this. The proof of Theorem 3.1 will occupy the whole of this section. We begin with a well-known lemma which follows directly from the relations in H Žsee, for example, w2, Lemma 3.3x.. LEMMA 3.2.
Suppose that 1 F i - n and that 1 F j F n. Then
Ža. L i commutes with L j . Žb. Ti commutes with L j if j / i, i q 1. Žc. Ti commutes with L i L iq1. The elements l 1 , . . . , l n also satisfy similar relations to those given in the lemma; however, we shall not need this. LEMMA 3.3. integers.
Suppose that 1 F i - n and let k and m be non-negati¨ e
Ža. If k G m then myc Lki Lmiy1Tiy1 s Tiy1 Lmi Lkiy1 q Ž q y 1 . Ý kcsmq1 Lci Lkq . iy1
Žb. If k F m then c kq myc . Lki Lmiy1Tiy1 s Tiy1 Lmi Lkiy1 y Ž q y 1 . Ý m cskq1 L i L iy1
Proof. Part Ža. is proved in w6, Lemma 5.10x; the argument is a routine induction on k y m Žuse Lemma 3.2Žc. to start the induction.. To prove Žb., apply the involution ) to Ža. and rearrange.
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Suppose that 1 - i F n and k G 1. Then
COROLLARY 3.4.
Lki s qy1 Tiy1 Lkiy1Tiy1 q qy1 Ž q y 1 .
ky1
Ý Lci Lkyc iy1 Tiy1 . cs1
Proof. When k s 1 this is just the definition of L i . In general, taking m s 0 in part Ža. of the lemma, we see that 1 Lkq s qy1 Lki Tiy1 L iy1Tiy1 i
½
s qy1 Tiy1 Lkiy1 q Ž q y 1 .
k
Ý Lci Lkyc iy1 cs1
y1 s qy1 Tiy1 Lkq1 Ž q y 1. iy1 Tiy1 q q
5
L iy1Tiy1
k
Tiy1 . Ý Lci Lkq1yc iy1 cs1
Recall the trace form f H : H ª A from Ž1.3.. Given h, hX g H we define fH
fH
an equivalence relation s on H by saying that h s hX whenever f H Ž h. s f H Ž hX .. This equivalence relation does not respect multiplication in H ; however, because f H is a trace form, it does have the following crucial property. fH
Ž3.5. Let h, hX g H . Then hhX s hX h. For j s 0, . . . , n let H j be the A-submodule of H with basis
Laj
j
??? La11 Tw N 0 F a i - r , w g S n 4 .
Note that Hn s H and that H j is not an H-module for j / n. However, H j is always a right H Ž S n , q .-module and, by Proposition 2.2, we also have the next result. Ž3.6. Suppose that 1 F j - n, 1 F k F r y 1 and h g Hj . Then fH
H Lkjq1 h s 0.
What we really need to know, however, is how to compute f H Ž Lrjq1 h.. LEMMA 3.7. Suppose that i, j, and k are integers such that 1 F i F j - n and 1 F k F r, and let h g H j . Then fH
Lkjq1 h s q iyjy1 Tj ??? Ti Lki Ti ??? Tj h.
283
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
Proof. For the duration of this proof let Tj??i s Tj ??? Ti and Ti?? j s Ti ??? Tj . By declaring that Tj??iq1 s 1 s Tiq1?? j we can state the lemma when i s j q 1; in which case the lemma asserts that f H Ž Lkjq1 h. s f H Ž Lkjq1 h.. Since this is certainly true, we may now proceed by downwards induction on i F j. Now, by induction and by Corollary 3.4, fH
q jyiq2 Lkjq1 h s qTj??i Lki Ti?? j h ky1
s Tj??i Tiy1 Lkiy1Tiy1Ti?? j h q Ž q y 1 .
Ý Tj??i Lci Lkyc iy1 Tiy1Ti?? j h. cs1 fH
Thus, it suffices to show that Tj??i Lci Lkyc iy1 Tiy1Ti?? j h s 0 whenever 0 - c - k. Fix such an integer c. Then, using the definition of the elements L a together with Lemma 3.2, we find c jyiq2 cy1 y1 y1 Tj??i Lci Lky Tj??i Lci Lky Tj??i L jq1 . h Ž Tiy1 iy1 Tiy1Ti?? j h s q iy1 cy1 c y1 y1 s q jyiq2 Tj??i Ž Lky L i . Tiy1 Tj??i L jq1 h iy1 cy1 1 y1 s q jyiq1 Tj??i Lky Tiy1 L iy1 . Tj??i L jq1 h Ž Lcy iy1 i 1 ky cy1 y1 s q jyiq1 Tj??i Lcy L iy1 Tiy1Tj??i L jq1 L iy1 h. i
Now Tay1 s qy1 ŽTa q 1 y q . for 1 F a - n. Therefore, by Lemma 3.3, together with Lemma 3.2Ža., there exist scalars a L g A such that 1 ky cy1 y1 Lcy L iy1 Tiy1Tj??i s i
Ý
jq 1 ??? L b 1 gB U LsTw Lbjq1 1
jq 1 ??? Lb 1 , a L Tw Lbjq1 1
where bjq1 F max c y 1, k y c y 14 F k y 2 whenever a L / 0. Consequently, c Tj??i Lci Lky iy1 Tiy1Ti?? j h s
jq 1 ??? Lb 1 L q jyiq1a L Tj??i Tw Lbjq1 1 jq1 L iy1 h
Ý jq 1 ??? L b1 LsTw Lbjq1 1
fH
s
jq 1 ??? Lb 1 L q jyiq1 a L L1qb jq1 1 iy1 hTj??i Tw
Ý jq 1 LsTw Lbjq1
???
Lb11 fH
by Ž3.5.. However, 1 q bjq1 F k y 1 - r; so Tj??i Lci Lkyc iy1 Tiy1Ti?? j h s 0 by Ž3.6. as required.
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MALLE AND MATHAS
Recall the element l jq1 s qyj Tj ??? T1T1 ??? Tj . Let j be an integer such that 1 F j - n and suppose
PROPOSITION 3.8. that h g H j . Then
fH
r
Lrjq1 h s Ž y1 . u1 ??? u r h l jq1 . Proof. By Lemma 3.7, f H Ž Lrjq1 h. s qyj f H ŽTj ??? T1T0r T1 ??? Tj h.. Therefore, since ŽT0 y u1 . ??? ŽT0 y u r . s 0, we may write f H Ž Lrjq1 h. as an A-linear combination of terms f H ŽTj ??? T1T0k T1 ??? Tj h. for k s 0, . . . , r y 1. However, by Lemma 3.7 and Ž3.6., if 0 - k - r then fH
fH
Tj ??? T1T0k T1 ??? Tj h s q j Lkjq1 h s 0. fH
Therefore, Lrjq1 h s Žy1. r qyj u1 ??? u r hTj ??? T1T1 ??? Tj s Žy1. r u1 ??? u r h l jq1 as required. We can now prove the main result of this section. Proof of Theorem 3.1. Recall the bases B and C of H and the Gram matrix GB C s Ž f H Ž bc .. b g B, c g C from the beginning of this section. An arbitrary element of B is of the form Lbnn ??? Lb11 Tw for some w g S n and some integers 0 F bi - r. In the matrix GB C order the elements of B lexicographically according to the exponents Ž bn , . . . , b1 ., and similarly for the elements of C. For j s 0, . . . , n let Bj s B l Hj and C j s C l HjU and define Gj to be the Gram matrix GB j C j s Ž f H Ž bc .. b g B j, c g C j . Then GB C s Gn and our ordering of the elements of B and C is such that the matrix Gj sits in the top left hand corner of the matrix Gjq1 for each 0 F j - n. Suppose that 0 F j - n and let a s Žy1. r u1 ??? u r . Then we claim that, as an r = r-block matrix, Gjq1 has the form
Gj
0
???
0
0
0 .. . 0
0 .. . 0
??? ? ?? ? ??
0 ? ?? )
a Gj
0
a Gj
)
???
) .. . )
0
.
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
285
d Ž c1 . d Ž c jq 1 . c 1 jq 1 ??? Lb 1 T jq 1 To see this let b s Lbjq1 ??? l jq1 L1 ??? Lcjq1 1 w and c s T¨ l 1 be arbitrary elements of Bjq1 and C jq1 , respectively. Then jq 1 ??? Lb 1 T T l d Ž c 1 . ??? l d Ž c jq 1 . Lc 1 ??? Lc jq 1 bc s Lbjq1 1 w ¨ 1 1 jq1 jq1
fH
jq 1 qc jq 1 Lb jqc j ??? Lb 1 qc 1 T T l d Ž c 1 . ??? l d Ž c j . l d Ž c jq 1 . s Lbjq1 j 1 w ¨ 1 j jq1
by Ž3.5. and Lemma 3.2Ža.. If bjq1 s c jq1 s 0 the assertion is just the definition of Gj Žnote that d Ž0. s 0.. If 0 - bjq1 q c jq1 - r then f H Ž bc . s 0 by Ž3.6.. Finally, if bjq1 q c jq1 s r then by Proposition 3.8 fH
r
d Ž c jq 1 . bc s Ž y1 . u1 ??? u r Lbj jqc j ??? Lb11qc1 Tw T¨ l 1d Ž c1 . ??? l jd Ž c j . l jq1 l jq1 r
s Ž y1 . u1 ??? u r Lbj jqc j ??? Lb11qc1 Tw T¨ l 1d Ž c1 . ??? l jd Ž c j . , since d Ž c jq1 . s y1 because c jq1 ) 0. This proves the claim. Therefore, det Gjq1 s "Ž u1 ??? u r . ry1 Ždet Gj . r for all 0 F j - n. Consequently, det GB C s det Gn s "Ž u1 ??? u r . m Ždet G0 . k for some integers k and m. However, G0 s Ž f H ŽTw T¨ .. ¨ , w g S n and it is well known, and easy to prove, that f H ŽTw T¨ . s q ldw ¨ y1 , where l is the length of a reduced expression for w. Hence, det G0 s "q z for some integer z. This completes the proof of Theorem 3.1.
4. THE DETERMINANT OF THE GRAM MATRIX OF TYPE GŽ r, 2, 2. For any divisor p of r let Wn, p be the normal subgroup of index p of Wn generated by the n q 1 reflections
t 0p , ty1 0 t 1 t 0 , t 1 , . . . , t ny1 4 if p - r, respectively, the n reflections
ty1 0 t 1 t 0 , t 1 , . . . , t ny1 4 if p s r. Thus Wn, p is the imprimitive complex reflection group GŽ r, p, n. in the notation of Shephard and Todd. We would like to introduce a symmetrizing form on the generic cyclotomic algebra H ŽWn, p , u. defined in w5x. This coincides with the algebra introduced by Ariki in w1x except for the case where n s 2, and both r and p are even, where the generic algebra has one additional parameter. In this latter case, no quasi-symmetric basis had been given in w3x, so we need to treat this algebra separately.
286
MALLE AND MATHAS
So, suppose that r is even, n s p s 2, and set d [ rr2. Then W2, 2 has a presentation described by the diagram
"
D Ž2d. :
Ž 4.1.
#
"
d "
Let H s H ŽW2, 2 , u., where u s Ž u1 , . . . , u d ; q; qX ., be the corresponding cyclotomic algebra, generated as associative unital Zwu, uy1 x-algebra by elements R, S, T subject to the relations
Ž T y u1 . ??? Ž T y u d . s 0 s Ž R y q . Ž R q 1 . s Ž S y qX . Ž S q 1 . , RST s STR s TRS Žsee w4, 4Ax.. We let U [ RS and V [ Ž qX y 1.UrqX q q y 1. Note that T and U commute and the element TU is central in H . The following result is inspired by w1, Lemma 1.2x. Let j, k be non-negati¨ e integers. Then
LEMMA 4.2.
Ž a. ST j s Ž qqX .
yj
T jU 2 j S y
1 q
jy1
T jUV
Ý Ž qqX . yiU 2 i ,
is0
Ž b. SU k s Ž qqX . Uyk S q k
1 q
ky1
UV
Ý Ž qqX . kyiU 2 iyk ,
is0
Ž c.
¡y 1 T V j
X
ST U s Ž qq . j
k
ky j
j
T U
2 jyk
S q~
q
1
¢q T V j
jyky1
Ý Ž qqX . yi U 2 iq1qk
is0 y1
Ý Ž qqX . yi U 2 iq1qk
if j G k, if j - k.
isjyk
Proof. We prove Ža. by induction on j. The result trivially holds if j s 0. If j s 1 then ST s STRRy1 s s Ž qqX .
y1
1 q
TUR y
TU 2 S y
1 q
qy1 q
TUV ,
TU s
1 q
TU 2 Sy1 y
qy1 q
TU
287
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
as claimed. The induction hypothesis and the case j s 1 now yield ST jq1 s Ž qqX . s Ž qqX .
yj
T jU 2 j ST y
yjy1
1 q
jy1
T jq1U
Ý Ž qqX . yiU 2 i V
is0
T jq1 U 2 jq2 S y
1 q
j
T jq1 U Ý Ž qqX .
yi
U 2iV ,
is0
proving the first part. In part Žb. the case k s 0 is again obvious. If k s 1 then SU s SRS s Ž qX Sy1 q qX y 1 .Ž qRy1 q q y 1 . S s qqX Uy1 S q Ž q y 1 . qX q Ž qX y 1 . Ž qRy1 q q y 1 . S s qqX Uy1 S q Ž q y 1 . qX q Ž qX y 1 . U s qqX Uy1 S q qX V . The inductive step is verified by the same type of calculation as in the first part. The last part is an immediate consequence of parts Ža. and Žb.. Note that T, U generate a commutative subalgebra of H . Since it is known that H is Zwu, uy1 x-free of rank < W2, 2 < by w4, Satz 4.7x we conclude as there that
T jU k , T jU k S N 0 F 2 j y k , k F r y 1 4 is a Zwu, uy1 x-basis of H Žcompare also with w1, Proposition 1.4x where the case q s qX is considered.. We proceed to show that this basis has good properties. PROPOSITION 4.3. The basis C s T jU k , T jU k S N 0 F 2 j y k, k F r y 1 4
Ž 4.4.
of H s H ŽW2, 2 , u. is quasi-symmetric. Proof. We have to show that the linear form f H : H ª Zwu, uy1 x defined by f H Ž1. s 1, f H Ž b . s 0 if 1 / b g C is a trace form. Since it specializes to the standard symmetrizing form of the complex reflection group W2, 2 the induced symmetric form on H is then automatically non-degenerate over the quotient field of A. We will have to consider elements of H of the form T jU k S l with 0 F k F r y 1, l g 0, 14 , and j an arbitrary integer. First note that on such elements fH Ž T jU k S l . s 0
if k / 0 or l / 0.
Ž 4.5.
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MALLE AND MATHAS
Indeed, if T jU k S l f C then by the relation on T we may replace T j by a linear combination of elements T jqi or T jyi, with 1 F i F d. Induction and the definition of f H now show the claim. Since R and S are invertible in H it clearly suffices to prove that f H Ž bL. s f H Ž Lb . for b g C and L g S, U, TU 4 . This is immediate for the central element L s TU. Now assume L s S. Inspection shows that if 0 F 2 j y k, k F r y 1 then all monomials on the right hand side of Lemma 4.2Žc. lie in C. Thus, Lemma 4.2Žc. gives the decomposition in C of ST jU k . By the definition of f H it follows that f H Ž S ? T jU k . s 0 s f H ŽT jU k ? S .. Similarly, f H ŽT jU k S ? S . equals fH Ž S ? T jU k S . s
½
qX 0
if j s k s 0, else.
So finally assume that L s U. As U commutes with T, the assertion is again clear if b s T jU k . For b s T jU k S we have to compare the value of f H on T jU kq 1S and on T jU k SU s qqX T jU ky 1S q qX Ž q y 1 . T jU k q Ž qX y 1 . T jU kq 1 Ž 4.6. Žby Lemma 4.2Žb... If k / 0, r y 1, and 2 j y k / 0, r y 1 then all monomials in the above expressions lie in C and f H vanishes on both sides. If 2 j y k g 0, r y 14 then f H again vanishes by Ž4.5.. We consider the two remaining cases separately. First assume that k s 0. We have f H ŽU ? T j S . s f H Ž ST jU . by what we proved for S, so the result is true if j s 0. For j ) 0 ST jU s TUST jy1 s Ž qqX .
1y j
T jU 2 jy1 S y
1 q
jy2
Ý Ž qqX . yiU 2 i
T jU 2 V
is0
by Lemma 4.2Ža.. All monomials appearing in the right-hand side are non-trivial basis elements, so f H vanishes. On the other hand we have the equality dy1
SU s S
Ý
dy1
a l T lq1U s
ls0 dy1
s
Ý ls0
Ý al TUST l ls0
a l Ž qqX .
ž
yl
T lq1 U 2 lq1 S y
1 q
ly1
T lq1 U 2 V Ý Ž qqX . is0
yi
U 2i
/
Žwith suitable constants a l . since TU is central and by Lemma 4.2Ža.. Here, f H is zero by Ž4.5..
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
289
It remains to consider k s r y 1. Lemma 4.2Ža. gives ST d s Ž qqX . y
yd
T d U r Ž S y Ž qX y 1 . .
qy1 q
T d U ry1 Ž qqX .
1y d
y
1 q
dy2
T d UV
Ý Ž qqX . yiU 2 i . Ž 4.7.
is0
The last two terms on the right-hand side are of the type Ž4.5.. On the other hand, replacing T d by a linear combination of T i with 0 F i F d y 1 and again applying Lemma 4.2Ža. shows that the left-hand side is a sum of elements of the form considered in Ž4.5.. Multiplication of both sides by TyŽ dyj. does not change this fact, and thus f H ŽT jU r S . s Ž qX y 1. f H ŽT jU r .. But now Ž4.6., with k s r y 1, gives f H Ž T jU ry1S ? U . s Ž qX y 1 . f H Ž T jU r . s f H Ž T jU r S . s f H Ž U ? T jU ry1S . as required. We write GC for the Gram matrix of f H with respect to the pair of bases C, CU , where CU is obtained from C by reversing the order of the factors in each element. PROPOSITION 4.8. Let C be the basis of H defined in Ž4.4.. Then detŽ GC . is in¨ ertible in A s Zwu, uy1 x. In particular H is symmetric o¨ er A. Proof. Elements of C are of the form T jU k S l for some j, k, l with 0 F k, 2 j y k F r y 1 and l g 0, 14 . We order C lexicographically with respect to Ž k, l, j .. Then Ž4.5. shows that the Gram matrix has the shape
G1 0 .. . 0
0 0 .. . 0
??? ??? ? ?? ? ??
0 0 ? ?? )
0
Gr
)
???
0 G2 ) .. . )
0
Žas an r = r-block matrix.. Again by Ž4.5. we see that G1 is a block diagonal 2 = 2-matrix, and the blocks on the diagonal are the same as for the algebra H ŽW1 , u.. Thus the determinant of G1 has the required form. The matrices Gi , 2 F i F r, can also be considered as 2 = 2-block matri-
290
MALLE AND MATHAS
ces, with entries f H ŽT j1qj 2 U r S l 1ql 2 . where l 1 , l 2 g 0, 14 . Application of Ž4.7. shows that Gi has the block form
ž
GiX
Ž qX y 1 . GiX
Ž qX y 1 . GiX
Ž qX q Ž qX y 1. 2 . GiX
/
.
Here GiX is the matrix with entries f H ŽT j1qj 2 U r .. Subtraction of Ž qX y 1. times the first block of r columns from the second block of r columns thus yields det Gi s qX detŽ GiX . 2 . To evaluate the determinant of GiX we need a further calculation. Application of Ž4.7. gives 1 T l U r s X T l U r Ž S y Ž qX y 1 . . S q s
1 q
X
d Ž qqX . T lyd ST d q
ž
qy1 q
T d U ry1 Ž qqX .
1yd
dy2 1 yi y T d UV Ý Ž qqX . U 2 i S q is0
/
s q Ž qqX .
dy 1
T lydST dS q Ž q y 1 . T l U ry1 S
dy2
y T l UV
Ý Ž qqX . yiy1U 2 iS.
is0
According to Ž4.5. the form f H vanishes on all summands on the right-hand 1 j side, except possibly the first. Hence, expanding T d s Ý dy js0 a j T , we have fH
T l U r s q Ž qqX .
dy 1
dy1
T lydS
Ý a jT jS js0
s q Ž qqX .
dy 1
dy1
T lyd
ž
js0 fH
s q Ž qqX .
dy 1
1
jy1
Ý a j Ž qqX . yj T jU 2 jS y q T jUV Ý Ž qqX . yiU 2 i fH
is0
/
S
T lyda0 S 2 s Ž qqX . u1 ??? u d T lyd d
by Lemma 4.2Ža. and Ž4.5.. Note that this is the analogue of Proposition 3.8 in this case. In particular this last equation shows that the determinant of GiX is the same as the one occurring in H ŽW1 , u. Žup to units in A.. The assertion follows. Let f H : H ª A be the symmetrizing form induced by the quasi-symmetric basis C. Over the quotient field K [ QuotŽ A. we may decompose f H as a sum fH s Ý dx x x gIrr Ž HK .
SYMMETRIC CYCLOTOMIC HECKE ALGEBRAS
291
of irreducible characters of HK [ H m K, with dx g K. The coefficients dx are called the relati¨ e degrees associated to f H , respectively, to the quasisymmetric basis C. Clearly we have
Ý
x gIrr Ž HK .
dx x Ž c . s 0
for all elements c / 1 of a quasi-symmetric basis with respect to f H , and Ý dx x Ž1. s 1. Using this property of the relative degrees the first author in w8x computed candidates for the relative degrees attached to a nice symmetrizing form on H . We can now verify: COROLLARY 4.9. The relati¨ e degrees of the symmetric form induced by the quasi-symmetric basis C are those gi¨ en in w8, Theorem 3.9x. Proof. According to the result in w8x it suffices to show that f H vanishes on the elements
RT jU j , ST jU j N 0 F j F d y 1 4 j T iqjU j N 0 - 2 i q j F r y 1 4 of H . But RT jU j s T jU jq1 Sy1 s Ž1rqX .T jU jq1 Ž S y Ž qX y 1.., on which f H vanishes by Ž4.7., ST jU j s T jU j S g C _ 14 , and also T iqjU j g C, so the result follows.
5. THE CYCLOTOMIC ALGEBRAS H Ž GŽ r, p, n., u. Finally, we consider the general case where Wn, p is an arbitrary imprimitive complex reflection group GŽ r, p, n.. Let H ŽWn, p , u. be the generic cyclotomic algebra associated to Wn, p in w5x. This coincides with the algebra defined by Ariki w1, Proposition 1.6x except in the case where n s 2 and r, p are both even. THEOREM 5.1. Let W s GŽ r, p, n. be an imprimiti¨ e complex reflection group. Then the associated generic cyclotomic algebra H s H ŽW, u. carries a canonical symmetrizing form f H o¨ er Zwu, uy1 x. This form ¨ anishes on all non-identity elements of the Ariki]Koike basis of H , while f H Ž1. s 1. In the case p s 1 it also ¨ anishes on all non-identity elements of any reduced basis of H . Proof. We reduce this to the case p s 1 Žrespectively p s 2 if n s 2 and r, p are even. where such a symmetrizing form was found in Theorem 3.1 Žresp. Proposition 4.8.. Assume first that either n ) 2 or one of r or p is odd. Let d [ rrp, z p [ expŽ2p irp. and A˜ [ Zw z p , u, uy1 x. Then according to Ariki w1, Proposition 1.6x the algebra H m A˜ can be identified
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˜ with a subalgebra of the specialized A-algebra H˜s H ŽWn , ˜ u., with parameters ˜u s Ž u1 , . . . , u d , z p u1 , . . . , z p u d , . . . , z ppy 1 u1 , . . . , z ppy 1 u d ; q . , Ž 5.2. generated by
T0p , Ty1 0 T1T0 , T1 , . . . , Tny1 4 . Let f˜ be the linear form on H˜ obtained by specializing the symmetrizing form constructed in Section 3. Since the specialization to the group algebra factorizes through the specialization Ž5.2. the form f˜ gives rise to a non-degenerate symmetric form on H˜. In particular, all irreducible charac˜ This implies that all ters of H˜m QŽu. occur with non-zero multiplicity in f. irreducible characters of the subalgebra H m A˜ of H˜ occur with non-zero ˜ Thus, fH is a symmetrizing multiplicity in the restriction f H of f˜ to H m A. ˜ form for H over the quotient field of A. It is shown in w1, Proposition 1.6x that
Lan
n
??? La11 Tw N 0 F a i F r y 1,
Ý ai ' 0 Ž mod p . , w g S n 4
˜ this gives a basis of H m A˜ which is is an A-basis of H . Tensoring with A, a subset of the Ariki]Koike basis of the cyclotomic algebra H˜ ; it follows ˜ However, the Gram determithat f H is a symmetrizing form for H m A. ˜ nants for the bases of H and H m A are equal and are elements of A, so f H is a symmetrizing form for H Žover A., as required. In the case where n s 2 and r and p are even, we embed H instead into a suitable specialization H ŽW2, 2 , ˜ u. of the generic algebra of W2, 2 studied Ž w in Section 4 see 8, Proposition 3.8x.. The remainder of the argument is then the same as in the general case. Remark 5.3. We expect that f H should also vanish on non-identity reduced basis elements in the general case p / 1. In this context note that a word which is reduced in Wn, p does not necessarily have to be reduced in Wn . Remark 5.4. In w7, Verm. 2.20 and Satz 5.13x the first author put forward a conjecture on the values of the relative degrees dx associated to a nice symmetrizing form on H ŽWn, p .. Using the calculations in w8, Theorem 3.4x and Corollary 4.9 it follows that this conjecture is correct if n s 2 for the symmetrizing form studied in this paper. We expect that in general the relative degrees attached to the symmetrizing form f H above are those predicted in w7x.
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Finally, suppose that O is an integral domain and let HO ŽWn, p , ˆ u. be the specialized cyclotomic Hecke algebra of Wn, p with parameters ˆ u g O. Given a set of indeterminates u over Z we can consider O as a Zwux-module by specifying that u acts as ˆ u. Then HO ŽWn, p , ˆ u. ( HZwuxŽWn, p , u. mZwux O . Therefore, by Theorem 5.1, we have the following. COROLLARY 5.5. Let HO ŽWn, p , ˆ u. be a specialized cyclotomic Hecke algebra o¨ er O . Then HO ŽWn, p , ˆ u. is a symmetric algebra with respect to f H if and only if the parameters ˆ u are in¨ ertible in O . In w6x an analogue of the q-Schur algebra for a cyclotomic Hecke alˆ gebra HO of type BnŽ r . was introduced. Key to the main results of that paper is the calculation of the ‘‘double annihilator’’ of certain elements of H Žsee w6, Theorem 5.16x.. When the parameters u are invertible elements of O , the required result of w6x is an immediate consequence of Corollary 5.5 Žsignificantly simplifying the argument there.. Interestingly, the results of w6x require only that qˆ be invertible in O . This suggests that HO may be a quasi-Frobenius algebra so long as qˆ is an invertible element of O .
REFERENCES 1. S. Ariki, Representation theory of a Hecke algebra of GŽ r, p, n., J. Algebra 177 Ž1995., 164]185. 2. S. Ariki and K. Koike, A Hecke algebra of ŽZrrZ. X Sn and construction of its irreducible representations, Ad¨ . in Math. 106 Ž1994., 216]243. 3. K. Bremke and G. Malle, Reduced words and a length function for GŽ e, 1, n., Indag. Math. 8 Ž1997., 453]469. 4. M. Broue 212 Ž1993., 119]189. ´ and G. Malle, Zyklotomische Heckealgebren, Asterisque ´ 5. M. Broue, ´ G. Malle, and R. Rouquier, On complex reflection groups and their associated braid groups, in ‘‘Representations of Groups’’ ŽB. N. Allison and G. H. Cliff, Eds.., CMS Conference Proceedings, Vol. 16, pp. 1]13, Amer. Math. Soc., Providence, 1995. 6. R. Dipper, G. James, and A. Mathas, Cyclotomic q-Schur algebras, Math. Z., in press. 7. G. Malle, Unipotente Grade imprimitiver komplexer Spiegelungsgruppen, J. Algebra 177 Ž1995., 768]826. 8. G. Malle, Degres cyclotomiques associees ´ relatifs des algebres ` ´ aux groupes de reflexions ´ complexes de dimension deux, in ‘‘Finite Reductive Groups: Related Structures and Representations’’ ŽM. Cabanes, Ed.., Progr. Math, Vol. 141, pp. 311]332, Birkhauser, ¨ Boston, 1997.