Invariant Algebra and Cuspidal Representations of Finite Monoids

Journal of Algebra 212, 721]737 Ž1999. Article ID jabr.1998.7634, available online at http:rrwww.idealibrary.com on

Invariant Algebra and Cuspidal Representations of Finite Monoids Mohan S. Putcha* Department of Mathematics, North Carolina State Uni¨ ersity, Raleigh, North Carolina 27695-8205 Communicated by Susan Montgomery Received July 22, 1997

Motivated by the theories of Hecke algebras and Schur algebras, we consider in this paper the algebra C M G of G-invariants of a finite monoid M with unit group G. If M is a regular ‘‘balanced’’ monoid, we show that C M G is a quasi-hereditary algebra. In such a case, we find the blocks of C M G to be the ‘‘sections’’ of the blocks of C M. We go on to develop a theory of cuspidal representations for balanced monoids. We then apply our results to the full transformation semigroup and the multiplicative monoid of triangular matrices over a finite field. Q 1999 Academic Press

INTRODUCTION Monoids and their representations arise naturally in connection with group representations. Here are a few examples. By w14x, unitary representations of the infinite symmetric group can be obtained from representations of finite symmetric inverse semigroups. Associated with a representation u of a reductive group G is the reductive monoid M Ž u . s u Ž G . , the Zariski closure of u Ž G . in the full linear monoid, cf. w15, 7x. For a finite Lie type group G and a modular irreducible representation u in the defining characteristic, the correct monoid M Ž u . is obtained in w20x, using some idempotents derived from the results of w21, 1x. For an irreducible complex representation u of the symmetric group Sn and the primitive idempotent * E-mail address: [email protected]. 721 0021-8693r99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.

722

MOHAN S. PUTCHA

e associated with the Young diagram as in w5; Section 28x, let J s S n eSn and M Ž u . s Sn j J j  04 . Then M Ž u . is a monoid and all idempotents of J are conjugate, a special property shared by reductive monoids. Because the theory of complex representations of finite groups is so well developed, we are motivated in w17, 18x to begin a systematic study of complex representations of finite monoids. Let M be a finite monoid with unit group G. By classical semigroup representation theory, the irreducible characters of M are in 1-1 correspondence with those of the maximal subgroups H J of regular J-classes J of M. The irreducible characters of the groups H J thus serve as ‘‘weights’’ for the irreducible characters of M in much the same way as in the representation theory of algebraic groups or Lie algebras. For an irreducible character u of H J we constructed in w17x, left induced, right induced, and semigroup induced characters uq, uy, and u˜ of G and we showed that u˜ is a summand of uql uy. Although the characters uq and uy of G can be expressed explicitly as sums of some induced group characters, the character u˜ is quite mysterious. For an algebra A with group G of automorphisms, the algebra AG of G-invariants is a natural and classical object of study, cf. w12x. In this paper we study the algebra C M G where M is a finite monoid with unit group G. We show that the invariant algebra C M JG of the local monoid M J is a quasi-hereditary algebra if and only if J is G-balanced; i.e., every component of uql uy is a component of u˜ for all irreducible characters u of H J . For G-balanced monoids we develop a theory of cuspidal characters, which enables us to classify the irreducible characters of G into series depending on which regular J-class J they come from, by generalized semigroup induction via a chain of regular J-classes. For the universal canonical monoid of Lie type, we have already shown in w13, 19x that these cuspidal characters are the same as those of Harish-Chandra w8x. Here we show that for the full transformation semigroup, only the alternating character is cuspidal. For the multiplicative monoid of triangular matrices over a finite field, we find that the components of the restriction of the Gelfand]Graev character Žof the general linear group. are all cuspidal. For regular G-balanced monoids, we show that the blocks of the invariant algebra are given by sections of the blocks of the original monoid algebra. We work this out for the full transformation semigroup of degree 4. In this case the invariant algebra is a 19-dimensional algebra of finite representation type, having 13 blocks.

723

INVARIANT ALGEBRA

1. PRELIMINARIES Let A be a finite-dimensional algebra over C with radical rad A. By a module we will mean a left module. We denote by Irr A and char A, the set of irreducible characters and character ring of A, respectively. The decomposition of A into a direct sum of indecomposable ideals Žor blocks. partitions Irr A into what we will also call blocks. The algebra A is called quasi-hereditary Žcf. w11x. if A has a chain of ideals, 0 s I0 ; I1 ; ??? ; It s A, such that for k s 1, . . . , t, Ži. Žii. Žiii.

IkrIky1 is a projective ArIky1 module Ik2 s Ik Ik ? rad A ? Ik : Iky1.

These algebras were introduced by Cline, Parshall, and Scott w4x in connection with the representation theory of Lie algebras and algebraic groups. They are exactly the finite-dimensional algebras whose modules form a ‘‘highest weight category.’’ Let M be a finite monoid with unit group G. We let C M denote the complex monoid algebra of M. We let Irr M s Irr C M, char M s char C M. If M has a zero, then C 0 M is the contracted monoid algebra, that is, the zero of M is the zero of C 0 M. If w , c are characters of G, then

ws

Ý

ugIrr G

mu u ,

cs

Ý

u gIrr G

nu u ,

mu G 0, nu G 0.

The intertwining number is defined by

Ž w, c . s Ž w, c .G s

Ý

ugIrr G

mu nu s

1
Ý wŽ g.c Ž g. . ggG

Let

wlcs

Ý

ugIrr G

pu u , where pu s min  mu , nu 4 .

If P is a subgroup of G, u is a character of P, then u ­G denotes the character of G induced from P into G,

u ­G Ž g . s

1 < P<

Ý

u Ž xgxy1 . .

xgG xgxy1 gP

If u is a character of G, then u x P denotes the restriction of u to P.

724

MOHAN S. PUTCHA

We let EŽ M . denote the idempotent set of M with the usual partial order, e F f , if ef s fe s e. If X : M, then EŽ X . s X l EŽ M .. The letters J, R, L , H will denote the usual Green’s relations on M: If a, b g M, then a J b if MaM s MbM, a R b if aM s bM, a L b if Ma s Mb, H s R l L . If J is a J-class, we call M J s G j J j  04 , a local monoid where for a, b g J, a( b s

½

ab, 0,

if ab g J . if ab f J

Then the span C J of J is an ideal of C 0 M J and is a ŽC M, C M .-bimodule. If J1 , J 2 are J-classes, then J1 F J 2 , if J1 : MJ2 M. A J-class J is regular if EŽ J . / B. Let U s U Ž M . denote the partially ordered set of regular J-classes of M. Then C Mrrad C M (

[U C Jrrad C J. Jg

For J g U Ž M ., choose e J g EŽ J .. Let H J denote the H-class of e J , that is the unit group of e J Me J . By semigroup representation theory w3; Chap. 5x, the irreducible characters of M are in 1-1 correspondence with the weights Irr H J , J g U . We write this as Irr M ( " Irr H J . Jg U

Ž 1.

Correspondingly for J g U , CJ s

[

ugIrr H J

C Ju ,

Ž 2.

with each C Ju being an ideal of C 0 M J and a ŽC M, C M .-bimodule. If u g Irr H J , then we denote by u˜ the character of G obtained by restricting to G the irreducible character of M associated with u . Let e be a primitive idempotent of C H J corresponding to u . Then DŽ u . s C J ? e ,

=Ž u . s Ž e ? C J . *

Ž 3.

are the standard and costandard C M-modules associated with u . As CG-modules, they yield characters uq and uy of G which have been

INVARIANT ALGEBRA

725

explicitly determined by the author w17x as sums of induced group characters. It is also shown in w17x that

u˜ is a summand of uql uy.

Ž 4.

If M is a regular monoid, that is a g aMa for all a g M, then with respect to the standard and costandard modules in the preceding text, the C Mmodules form a highest weight category, cf. w18x. In particular C M is a quasi-hereditary algebra. This was noted independently in w10, 18x. For any finite monoid M and regular J-class J, M J is a regular monoid, whereby C M J and C 0 M J are quasi-hereditary algebras. Let M be a finite regular monoid J, J9 g U , J9 ) J, u g Irr H J . Let L s  J9 s J 0 ) ??? ) Jt 4 be a chain of J-classes with Jt ) J. Let Hi s H J i , e i s e J i , i s 0, . . . , t. Let u t s uqy u˜g char Ht in the monoid e t Me t . Let u iy1 s u˜i g char Hiy1. Let aLqŽ u . s u 0 g char H0 . Similarly starting with Ž . u t s uyy u˜g char Ht , we get ay L u s u 0 g char H 0 . Let

bq J9 Ž u . s

Ý Ž y1. l L aLq Ž u . ,

by J9 Ž u . s

Ý Ž y1. l L aLy Ž u . ,

Ž .

L

Ž .

L

where the summations are over the chains of J-classes L s  J9 s J 0 ) ??? ) Jt 4 with Jt ) J. If u 9 g Irr H J 9 , define

u ª u 9, if Ž u 9, bq J 9 Ž u . . ) 0, u 9 ª u , if Ž u 9, by J Ž u . . ) 0.

Ž 5.

We have shown in w18x that the components of Ž Irr M, ª . are the blocks.

Ž 6.

Moreover, ŽIrr M, ª. is related to the Žordinary. quiver of C M. See w18x for details.

2. THE ALGEBRA OF INVARIANTS Let M be a finite monoid with unit group G. Then G acts on M and hence C M by conjugation. The algebra of in¨ ariants, C M G s  a g C M < ag s ga for all g g G 4 . We have studied this algebra in w16, 17x. If J g U , let C J G s  a g C J < ag s ga for all g g G 4 .

726

MOHAN S. PUTCHA

When C M is semisimple, we have in w16x explored the connections between C J G and Hecke algebras. For p g Irr G, let jp denote the associated primitive central idempotent of CG. Then clearly, CMG s

[

pgIrr G

jp ? C M.

Ž 7.

If J g U , then by Ž2. and Ž7., CJG s

[

pgIrr G

jp ? C J G s

[

p gIrr G u gIrr H J

jp ? C JuG ,

Ž 8.

with each jp C JuG being a ŽC M G , C M G . bimodule. Hence by Ž1. and Ž8., Irr C M G ( " Irr G = Irr H J . Jg U

Ž 9.

Let J g U , u g Irr H J . By w17, Theorem 1.4x, dim C JuG s Ž uq, uy . .

Ž 10 .

Let us elaborate. Let e be a primitive idempotent of C H J corresponding to u . Let e be an idempotent of C JuG such that e becomes the identity element modulo the radical of C JuG. Then ab s aeb, for all a, b g C Ju .

Ž 11 .

As left CG-modules, let e ? C Ju ? e s V1 [ ??? [ Vp , C Ju ? e s V1 [ ??? [ Vm

Ž 12 .

be decompositions into irreducible components. If x i is the character of the irreducible left CG-module Vi , then

u˜s x 1 q ??? qx p , uqs x 1 ??? qxm . As right CG-modules, let

e ? C Ju ? e s W1 [ ??? [ Wp , e ? C Ju s W1 [ ??? [ Wn

Ž 13 .

727

INVARIANT ALGEBRA

be decompositions into irreducible components. If c j is the character of the irreducible right CG-module Wj , then

u˜s c 1 q ??? qcp ,

Ž 14 .

uys c 1 ??? qcn . We may further assume that

x i s c i , for 1 F i F p.

Ž 15 .

Let Ui j s ² Vi Wj : ( Vi mC Wj . Then as vector spaces, C Ju s C JuG s

[U , i, j

ij

Ž 16 .

[U i, j

G ij ,

and Ui Gj (

½

C,

if x i s c j

0,

if x i / c j

.

Ž 17 .

For j s 1, . . . , n, let m

Yj s

[U . is1

ij

Then as left C M and left C M G-modules, respectively, n

C Ju s

[Y , js1

j

Ž 18 .

n

C JuG

s

[Y js1

j

G

.

Moreover as left C M G-modules, Yj G ( Yj9G

m

c j s c j9 .

Ž 19 .

By Ž11. ] Ž17., 2 Ž C Ju . s [ ½ Ui j < x i s c j is a component of u˜5 .

Ž 20 .

728

MOHAN S. PUTCHA

We say that a regular J-class J is a G-balanced if every component of uql uy is a component of u˜ Žcompare with Ž4... THEOREM 2.1. The following conditions are equi¨ alent. Ži. C 0 M JG is a quasi-hereditary algebra. Žii. J is G-balanced. Žiii. ŽC J G . 2 s C J G . Proof. That Žii. m Žiii. follows from Ž20.. We now prove Žii. m Ži.. Suppose J is G-balanced. It suffices to show that C J G is a projective C M G -module. Let u g Irr H J . By Ž18., it suffices to show that each Yj G is G q y a projective C G M -module. If Yj / 0, then c j is component of u l u . ˜ Ž Hence c j is a component of u . So c j s c i for some i F p. By 19., Yj G ( Yi G as left C M G modules. Now C Jurrad C Ju ( e ? C Ju ? e is a simple algebra. It follows from Ž16., that Yi G s C JuG ? f for some idempotent f F e. Hence Yi G is a projective C M G-module. Conversely assume that C M JG is a quasi-hereditary algebra. Let u g Irr H J . Let p be a component of uql uy and let j s jp be as in Ž7.. Let Z denote the center of CG and A s Z q C JuG . Then A is a homomorphic image of C 0 M JG. Let R denote the radical of C Ju . Then R G s rad A. Suppose p is not a component of u˜. Then by Ž13. ] Ž17., j R G / 0 and

j A ( C q j RG . Hence j A has no nontrivial idempotent ideals. Because j A is a homomorphic image of C 0 M JG , this contradicts the assumption that C 0 M JG is a quasi-hereditary algebra. Hence J is G-balanced, completing the proof. We say that M is G-balanced if J is G-balanced for all J g U . We say that M is balanced if for all e g EŽ M ., eMe is H-balanced, where H is the unit group of eMe. Let M be a regular monoid. Let Žp , u ., Žp 9, u 9. g Irr C M G as in Ž9.. Define Žp , u . ª Žp 9, u 9. if

u ª u 9, as in Ž 5 .

and

p s p 9 is a component of u˜l u˜9. Ž 21 .

For p g Irr G, define the p-section of ŽIrr M, ª. to be the subgraph consisting of u g Irr M with p a component of u˜. By Ž6., Ž7., and

729

INVARIANT ALGEBRA

Theorem 2.1, we have, THEOREM 2.2.

Suppose M is a G-balanced regular monoid. Then,

Ži. C M G is a quasi-hereditary algebra. Žii. The connected components of ŽIrr C M G , ª. are the blocks. Žiii. ŽIrr C M G , ª. is the disjoint union of the p-sections of ŽIrr M, ª., p g Irr G. Živ. C M G is semisimple if and only if the p-sections of ŽIrr M, ª. are discrete graphs for all p g Irr G. EXAMPLE 2.3. Let M s  1, s , e, e s , s e, s e s 4 subject to

s 2 s 1,

e 2 s e s e s e.

Then G s  1, s 4 , U s  G, J 4 , J s  e, e s , s e, s e s 4 . Then M is not G-balanced. The sections of ŽIrr M, ª. are discrete, but C M G is not semisimple.

3. CUSPIDAL CHARACTERS Let M be a monoid with unit group G and let G be a chain of idempotents in M. In much of our work, beginning with w15x, the opposite ‘‘parabolic subgroups,’’ P Ž G . s  x g G < xe s exe for all e g G 4 , P Ž G . s  x g G < ex s exe for all e g G 4 y

Ž 22 .

have played an important role. For reductive monoids they are exactly the opposite parabolic subgroups Žin the group theoretic sense. of the reductive group G, cf. w15x. For finite monoids M with C M semisimple we have in w13x Žwith Okninski. used these parabolic subgroups to develop a theory of cuspidal representations, which classifies the irreducible characters of G into series depending on which J-class they ‘‘come from.’’ Because C M is not semisimple for most finite monoids M, there is a need to generalize this theory. For this purpose, we begin by combining the two definitions of Ž22. into one. Let G s  1 s e0 ) ??? ) e t 4 be a chain of idempotents. For G9 : G, let P Ž G, G9 . s  x g G < e i xe i s e iy1 xe i if e i f G9, e i xe i s e i xe iy1 if e i g G9, i s 1, . . . , t 4 . Clearly P Ž G, B. s P Ž G . and P Ž G, G . s PyŽ G ..

730

MOHAN S. PUTCHA

LEMMA 3.1. P Ž G, G9. is a subgroup of G and e i xye i s e i xe i ye i for all x, y g P Ž G, G9., i s 0, . . . , t. Proof. We prove closure for i s 0, . . . , t, by induction on i. For i s 0, this is obvious. So let i ) 0. Let x, y g P Ž G, G9.. Suppose e i g G9. Then e i xye iy1 s e i e iy1 xye iy1 s e i e iy1 xe iy1 ye iy1 s e i xe iy1 ye iy1 s e i xe i ye iy1 s e i xe i ye i . Hence, e i xye iy1 s e i xye i s e i xe i ye i . Similarly if e i f G9, then e iy1 xye i s e i xye i s e i xe i ye i . In particular we have a homomorphism w : P Ž G, G9. ª H Ž e t . given by w Ž x . s e t xe t . Let J denote the J-class of e t . If u g Irr H J , then we have correspondingly u 9 g Irr H Ž e t .. Now u 9 lifts via u to a character u 0 of P Ž G, G9.. Let

u Ž G, G9 . s u 0­ G

Ž 23 .

denote the character of G induced from P Ž G, G9. to G. Now let L s  G s J 0 ) ??? ) Jt s J 4 be a chain of regular J-classes. Let u g Irr H J . We define the semigroup induced character u G of G as follows. Assume without loss of generality that 1 s e J 0 ) e J1 ) ??? ) e J t s e J . Let u t s u . Having obtained u i g char H J i , i ) 0, let u iy1 s u˜i g char H J iy 1 in the monoid e J iy 1 Me J iy 1. Let

u L s u 0 g char G.

Ž 24 .

Next let L9 : L. Then we construct a character u Ž L, L9. of G as follows. Let u t s u . Having obtained u i g char H J i , i ) 0, then in the monoid e J iy 1 Me J iy 1, let

u iy1 s

½

uq i , uy i ,

if Ji f L9 . if Ji g L9

Let

u Ž L , L9 . s u 0 g char G.

Ž 25 .

731

INVARIANT ALGEBRA

We wish to describe the character u Ž L, L9. in terms of the induced group characters in Ž23.. Let C s C Ž L, L9. denote the set of all pairs Ž G, G9. where G s  1 s e0 ) ??? ) e t 4 is a chain of idempotents with e i g Ji , i s 0, . . . , t, and G9 s  e i < Ji g L94 . Define an equivalence relation ; on C as follows. Let Ž G1 , G1X ., Ž G2 , G2X . g C . If G1 s  1 s e0 ) ??? ) e t 4 , G2 s  1 s f 0 ) ??? ) f t 4 , then Ž G1 , G1X . ; Ž G2 , G2X . if for i s 0, . . . , t, e i R f i , if Ji f L9,

e i L f i , if Ji g L9.

Clearly ; is an equivalence relation on C . Let w G, G9x denote the equivalence class of Ž G, G9.. Let u g Irr H J . We note that P Ž G, G9. and u Ž G, G9. depend only on w G, G9x. Now G acts by conjugation on Cr; . Let w G1 , GiX x, . . . , w Gp , GpX x denote the orbit representatives. Then p THEOREM 3.2. Ži. u Ž L, L9. s Ý is1 u Ž Gi , GiX . Žii. u L is a summand of u Ž L, L9. for all L9 : L.

Proof. Because Žii. follows from Ž4., we prove Ži.. We proceed by induction on < M <. Let e s e J1, H s H J1. By symmetry suppose that J1 f L9. Let P s P Ž e .. Let d : P ª H be the homomorphism defined by d Ž x . s xe, and let P9 s d Ž P .. Let LX1 s  J l eMe < J g L9 4 .

L 1 s  J l eMe < J g L 4 ,

Let w s u Ž L 1 , LX1 . g char H. Let w V 1 , VX1 x, . . . , w V s , VXs x denote the H-orbit representatives of C Ž L 1 , LX1 .r; . Let Pj s P Ž V j , VXj ., let u j be the associated character of Pj . Then by the induction hypothesis, s

ws

Ý u j ­ H. js1

Let w V j1 , VXj1 x, . . . , w V j p , VXj p x denote the orbit representatives of the action of PjX s P9 l Pj by conjugation on As

½

h

V j , VXj

hgH .

5

Let Pji s P Ž V ji , VXji ., let u ji be the associated character of Pji , PjiX s P9 l Pji . Then by Mackey’s subgroup formula w6; Theorem 10.13x, s

w x P9 s

Ý Ž u j ­ H . x PjX js1 s

s

p

Ý Ý Ž u ji x PjiX . ­ PjX . js1 is1

732

MOHAN S. PUTCHA

Lifting via d and inducing to G,

w(d s

Ý u Ž Gi , GiX . . egGi

Summing over the G-orbits of the R-classes of idempotents of J, we obtain the result. We are now in a position to generalize the theory of cuspidal characters in w13x. For e g EŽ M ., let U Ž e . s  x g G < xe s e 4 , Uy Ž e . s  x g G < ex s e 4 . Then UŽ e . 1 P Ž e . and UyŽ e . 1 PyŽ e .. Call an irreducible character p of G, M-cuspidal if for all J g U , J / G, u g Irr H J , p is not a component of u˜. THEOREM 3.3. Ži.

Let p g Irr G. Then

Suppose that for all J g U , e, e9 g EŽ J .,

p Ž u. s 0

Ý

or

ugU Ž e .

Ý

p Ž u . s 0.

ugU yŽ e9.

Then p is M-cuspidal. Žii. There is a chain L s  G s J 0 ) ??? ) Jt s J 4 in U and an irreducible e J Me J-cuspidal character u of H J such that p is a component of u L . Žiii. If M is balanced, then the components of u L are the same as the components of FL9: L u Ž L, L9.. Proof. Žii. follows by induction and Žiii. follows from Theorem 3.2. So we prove Ži.. Suppose p is not M-cuspidal. Then there exists J g U , J / G, u g Irr H J such that p is a component of u˜. Then by Ž4., p is a component of uql uy. Then for some e1 , e2 g EŽ J ., p is a component of u 1 ­G and u 2 ­G where u 1 , u 2 are characters of P Ž e1 . and PyŽ e2 ., associated with u . Then UŽ e1 . is contained in the kernel of u 1 and UyŽ e2 . is contained in the kernel of u 2 . By Frobenius reciprocity w6; Theorem 10.9x, u 1 is a summand of p x P Ž e1 . and u 2 is a summand of p x PyŽ e2 .. It follows that

Ý ugU Ž e 1 .

p Ž u . / 0 and

Ý

ugU yŽ e 2 .

p Ž u . / 0.

INVARIANT ALGEBRA

733

4. EXAMPLES Our first example is the full transformation semigroup Tn of all self-maps of n s  1, 2, . . . , n4 . It plays a pivotal role in semigroup and automata theory. Let k F n, a & k be a partition of k, let w a x be the associated irreducible character of the symmetric group Sk . If b & n y k, then the outer product w a xw b x is obtained by inducing the outer tensor product character Sk = Snyk ¨ S n to Sn . In particular w a xw n y k x is explicitly determined by Young’s rule w9; Corollary 2.8.3x. We have shown w17; Theorem 2.1x that in Tn , w a ˜ x s w a xqlw a xy, q

w a x s w a xw n y k x ,

Ž 26 .

and

w a˜ x s

½

w a xw n y k x , w n y k q 1, 1ky 1 x ,

if w a x / w 1k x , if w a x s w 1k x .

Ž 27 .

The character w a xy is much more difficult to determine. It is related to plethysms of characters, cf. w9, Section 5.4x. For a positive integer m, consider N s NS m kŽ Sm = ??? = Sm . with k copies of S m . Then NrSm = ??? = Sm ( Sk . So w a x lifts to a character of N which can then be induced to a character w m x ( w a x of Sm k , called the plethysm of w m x and w a x. Even though some partial results are known, the general problem of decomposing plethysms even for w a x s w k x, remains a difficult open problem in algebraic combinatorics. Let Jk denote the rank k J-class of Tn . The Sn = Sn orbits are in 1-1 correspondence with the partitions l s Ž n1k 1 , . . . , n kt t . with n1 ) ??? ) n t and k s k 1 q ??? qk t parts. Within this orbit, all idempotents are conjugate. Consider a typical such idempotent el. Then Uly s Uy Ž el . s Sl , Ply s Py Ž el . s NS nŽ Sl . , and if g s Ž k 1 , . . . , k t ., then PlyrUly s Sg : Sk . Let

w a x x Sg s Ý ma 1 , . . . , a t w a 1 x a ??? a w a t x , a i&k i

734

MOHAN S. PUTCHA

where a denotes the outer tensor product of characters. Now w a xx Sg lifts to a character w a x l of Ply . Clearly,

w a x l ­ Sn s Ý ma 1 , . . . , a tŽ w n1 x ( w a 1 x . ??? Ž w n t x ( w a t x . .

Ž 28 .

a i&k i

By w17x, y

w a x s Ý w a x l ­ Sn .

Ž 29 .

l

THEOREM 4.1.

Let 1 F k F n, w a x g Irr Sk . Then

Ži. w a xqs w a xw n y k x Žii. y wax s

Ý Ž w a x , w a 1 x ??? w a t x .

Ý

Ž n 1k1 , . . . , n kt t .&n , k 1 q ? ? ? qk tsk a i&k i

= Ž w n1 x ( w a 1 x . ??? Ž w n t x ( w a t x . Žiii. Živ. Žv. Žvi. Žvii.

wa ˜ x s w a xqlw a xys w a xw n y k x if w a x / w1k x. wa ˜ x s w a xqlw a xys w n y k q 1, 1ky 1 x if w a x s w1k x Tn is a balanced monoid. TnS n is a quasi-hereditary algebra. CT The alternating character w1n x is the only Tn-cuspidal character of

Sn . Proof. Ži., Žiii., and Živ. follow from Ž26. and Ž27.. Žii. follows from Ž28., Ž29., and Frobenius reciprocity. Žv. follows from Žiii. and Živ. because if e is Tn e ( Tk . Žvi. now follows from an idempotent in Tn of rank k, then eT Theorem 2.2. If k - n, then by Žiii. and Živ., w1n x is not a component of w a ˜ x for w a x g Irr S k . It follows that w1n x is a Tn-cuspidal character of Sn . Let w g x s w n1k 1 , . . . , n kt t x g Irr Sn , n1 ) 1. Then w g x is component of w b˜ x where w b x s w n1ky 1, n1 y 1, . . . , n kt t x. Hence w g x is not Sn-cuspidal. This proves Žvii.. COROLLARY 4.2. y

wkx s

For 1 F k F n,

Ý Ž n 1k1 , . . . , n kt t .&n k 1q ??? qk tsk

Ž w n1 x ( w k 1 x . ??? Ž w n t x ( w k t x . .

735

INVARIANT ALGEBRA

T2S 2 , CT T3S 3 are semisimple. So consider the case n s 4. By w18x, CT ŽIrr T4 , ª. is the ordinary quiver of CT T4 and is as follows, w1x ª w12 x ª w13 x ª w14 x ­ w3x

­ w3, 1x

­

x

w2 2 x

w2, 1x

w2, 12 x .

­ w4x x w2x The w4x-section is w4x ªw2x w1x w3x. The w3, 1x-section is w3x ª w12 x,

w3, 1x ª w2, 1x w2x.

The w2 2 x-section is

w 2 x w 2, 1 x w 2 2 x . The w2, 12 x-section is

w 2, 1 x w 2, 12 x w 13 x . The w14 x-section is

w 14 x . Hence we have, T4S 4 is a 19-dimensional algebra of finite representation COROLLARY 4.3. CT type ha¨ ing 13 blocks.

736

MOHAN S. PUTCHA

Our second example is the multiplicative monoid Tn s TnŽFq . of all upper triangular matrices over a finite field Fq . Let TnU denote the unit group of Tn . If J is a regular J-class of Tn , u g Irr H J , then by w17; Theorem 3.6x,

u˜s uql uy.

Ž 30 .

Let U denote the unipotent group of all upper triangular matrices with 1’s on the diagonal. For i - j, let the root subgroup, X i j s  I q a Ei j < a g Fq 4 . Then Us

Ł X i j , in any order. i-j

The commutator subgroup, U9 s

Ł

iq1-j

Xi j ,

and UrU9 s X 12 = X 23 = ??? = X ny1, n . Let d i g Irr X i, iq1 , p i / 1, d s d 1ad 2 a ??? g Irr UrU9. Let dˆ denote the lift of d to Irr U. The induced character G s dˆ ­GLnŽFq . is called the Gelfand]Grae¨ character of GLnŽFq ., cf. w2; Section 8.1x. THEOREM 4.4. Ži. Tn s TnŽFq . is a balanced monoid. Žii. If G is the Gelfand]Grae¨ character of GLnŽFq ., then all components of G xTnU are Tn-cuspidal. Proof. Ži. If e g EŽTn ., then eTn e s Tm 1 = ??? = Tm k for some m1 , . . . , m k . It follows from Ž30. that Tn is a balanced monoid. Žii. Let J be a regular J-class of Tn , J / TnU . Then by w17; Theorem 3.6x, J has a diagonal idempotent e and all idempotents of J are conjugate in Tn . By Theorem 3.3Ži., Frobenius reciprocity, and Mackey’s subgroup formula, it suffices to show that for some i, X i, iq1 is contained in UŽ e . or UyŽ e .. If the Ž1, 1. entry of e is 0, then X 12 : UyŽ e .. Otherwise for some i ) 1, the Ž i, i . entry of e is 0. Then X iy1 i : UŽ e .. This completes the proof. Conjecture 4.5. Let G be a finite group of Lie type and let B be a Borel subgroup of G. Let M denote the universal canonical monoid of

INVARIANT ALGEBRA

737

w19x. Let B denote the ‘‘triangular’’ monoid generated by B and the ‘‘diagonal’’ idempotents. Then the irreducible B-cuspidal characters of B are exactly the components of G x B where G is the Gelfand]Graev character of G.

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