Complex Representations of Finite Monoids II. Highest Weight Categories and Quivers

205, 53]76 Ž1998. JA977395

JOURNAL OF ALGEBRA ARTICLE NO.

Complex Representations of Finite Monoids II. Highest Weight Categories and Quivers Mohan S. PutchaU Department of Mathematics, North Carolina State Uni¨ ersity, Raleigh, North Carolina 27695-8205 E-mail: [email protected] Communicated by T. E. Hall Received November 28, 1996

In this paper we continue our study of complex representations of finite monoids. We begin by showing that the complex algebra of a finite regular monoid is a quasi-hereditary algebra and we identify the standard and costandard modules. We define the concept of a monoid quiver and compute it in terms of the group characters of the standard and costandard modules. We use our results to determine the blocks of the complex algebra of the full transformation semigroup. We show that there are only two blocks when the degree / 3. We also show that when the degree G 5, the complex algebra of the full transformation semigroup is not of finite representation type, answering negatively a conjecture of Ponizovskii. Q 1998 Academic Press

INTRODUCTION Semigroup theory has been extensively developed for well over half a century. We refer to w7x for current trends. The author has focused his work on connections with classical algebra; see, for example, w14, 16, 17, 19x. Here we continue from w18x our study of complex representations of finite monoids M. We restrict our attention to regular monoids Ž a g aMa for all a g M .. We remark that much of semigroup theory concerns regular semigroups. Let U denote the partially ordered set of J-classes of M and choose idempotents e J Ž J g U .. Let H J denote the unit group of e J Me J . By classical semigroup representation theory w3, Chap. 5x, the irreducible representations of M are indexed by L s DJ g U Irr H J . For U

Research partially supported by NSF. 53 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

54

MOHAN S. PUTCHA

the purposes of this paper, we think of L as the set of weights. For u g L we define the standard module DŽ u . and costandard module =Ž u . of the complex monoid algebra of C M. We then show that the C M-modules form a highest weight category in the sense of Cline et al. w4x. In particular C M is a quasi-hereditary algebra with global dimension bounded by twice the length of a longest chain in U Žthis has also been noted by Kuzmanovich and Teply.. If G is the unit group of M and u g L, then we computed in w18x the left and right induced characters uq and uy of the CG-modules DŽ u ., =Ž u ., respectively. Here we construct some new characters as alternating sums of induced characters via chains of J-classes. We use these characters to introduce the concept of a monoid quiver of M, closely related to the ordinary quiver of C M. We apply our results to the full transformation semigroup Tn of degree n. This is the regular monoid Žunder composition. of all self maps of  1, . . . , n4 . In this case L consists of all partitions of m, m F n. By Tn is not of finite considering the monoid quiver of Tn we show that CT representation type for n G 5, answering negatively a conjecture of Ponizovskii w15x. Tn . We show that for n / 3, CT Tn We go on to determine the blocks of CT has only two blocks with one of them being a simple algebra. The isolated block corresponds to w2, 1ny 2 x. For n s 3, both w2x and w2, 1x are isolated.

1. PRELIMINARIES Let M be a finite monoid with unit group G. For X : M, let EŽ X . denote the set of idempotents in X. 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 e g EŽ M ., then the H-class H s H Ž e . of e is the unit group of eMe. We will assume throughout that M is regular, whereby EŽ J . / B for all J-classes J. Let U denote the set of all J-classes of M. If J1 , J 2 g U , then J1 F J 2

if J1 : MJ2 M.

If e1 , e2 g EŽ M ., then e1 F e 2

if e1 s e1 e2 s e2 e1 .

Let J1 F J 2 in U . Then it is easy to see that e 2 g E Ž J 2 . « e1 F e 2

for some e1 g E Ž J1 . .

Ž 1.

COMPLEX REPRESENTATIONS

55

By an ideal of M we mean a subset Žpossibly empty. I such that MIM s I. If J g U , then I X Ž J . s MJM _ J

I Ž J . s MJM,

Ž 2.

are ideals of M. Let J 0 s J j  04 , where for a, b g J, a( b s

½

ab 0

if ab g J otherwise.

We let C M denote the complex monoid algebra of M, cf. w13x. If I is an ideal of M, then its span C I is an ideal of C M. If J g U , then we let C J denote the contracted semigroup algebra C 0 J 0 , i.e., the zero of C J is the zero of J 0 . For any algebra A, we let R s rad A denote the radical of A. By an A-module, we will mean a finitely generated left A-module. We will now briefly review classical semigroup representation theory w12; 2, Chap. 5x. Clearly linear complex representations of C M are in 1]1 correspondence with the C M-modules. Let R s rad C M. Then C MrR (

[U C Jrrad C J.

Ž 3.

Jg

Hence the irreducible representations of C M are in 1]1 correspondence with those of C J, J g U . Given an irreducible representation w of C M, the associated J-class J Žcalled the apex of w . is characterized by Ži. Žii.

wŽ J . / 0 w Ž J X . s 0 for J X g U , J X h J.

Let J g U , e g EŽ J ., H s H Ž e .. If J has m R-classes and n L-classes, then J 0 is a Rees matrix semigroup over H 0 s H j  04 with sandwich matrix P. P is an n = m matrix over H 0 and J 0 can be identified with m = n monomial matrices over H 0 with multiplication twisted as A( B s APB.

Ž 4.

Then C J is a Munn algebra over C H with sandwich matrix P. Thus C J can be identified with the algebra Žwithout identity. of m = n matrices over C H with multiplication given by Ž4.. To understand the structure of C J, consider first the algebra A of m = n matrices over C with multiplication given by Ž4. where P is an n = m complex matrix. Replacing P by an equivalent matrix leads to an isomorphic algebra. So if P has rank r, we can without loss of generality,

56

MOHAN S. PUTCHA

let P s w I0r 00 x. The map w : A ª Mr ŽC., the algebra of r = r matrices over C, given by

w

A C

B sA D

is a surjective homomorphism. Hence rad A s

½

A C

B As0 D

5

Ž 5.

and Arrad A ( Mr Ž C . .

Ž 6.

Also rad 2 A s

½

A C

B AsBsCs0 . D

5

Ž 7.

Now consider the algebra C J. C H is a direct sum of simple algebras. Correspondingly C J is a direct sum of algebras of the type discussed above. Hence by Ž6., the irreducible representations of C J are in 1]1 correspondence with those of C H. This is due to Clifford w2x. In particular a semisimple C J-module V is uniquely determined by the C H-module eV. By Ž5., we get C J ? rad C J ? C J s 0

Ž 8.

as noted in w9x. Also by Ž7., rad 2 C J s  a g C J N ax s xa s 0 for all x g J 4 .

Ž 9.

Let Irr G, char G denote the set of irreducible characters and the ring of virtual characters of G, respectively. Let P be a subgroup of G. If u g char P, then the Žvirtual. character induced from P to G,

u ­G Ž g . s

1 < P<

Ý

u Ž xgxy1 . .

xgG xgxy1 gP

If u g char G, then u x P g char P is the restriction of u to P. If u , c g char G, let Ž u , c . denote the intertwining number, cf. w5x. For J g U , choose e J g EŽ J . and let H J s H Ž e J .. Let L s LŽ M . s

" Irr HJ .

Jg U

Ž 10 .

COMPLEX REPRESENTATIONS

57

For u g Irr H J , let LŽ u . denote the associated irreducible C M-module and let u˜ denote the character of the CG-module LŽ u .. The map u ª u˜ extends to char H J . We note that changing e J changes H J by a naturally isomorphic group. So the particular choice of e J is not relevant. Let J g U , I s I Ž J . as in Ž2.. Choose hJ g C I such that

hJ is the unity of C I modulo R .

Ž 11 .

Let

j J s hJ ?

Ł Ž 1 y hJ . X

X

J -J

Ž 12 .

in some order. Modulo R, hJ , j J are central idempotents and j J is independent of order. If c : C M ª C MrR is the natural homomorphism, then

c Ž j J ? C I . ( C Jrrad C J.

Ž 13 .

Let V be a C M-module. Let G s  J 0 ) J1 ) ??? ) Jt 4 be a chain of J-classes. The length l Ž G . s t. Let e i s e J i , Hi s H J i . Because of Ž1., we can assume without loss of generality that e0 ) e1 ) ??? ) e t . Let u s u t denote the character of the C Ht-module e t V. Inductively, having obtained character u i of Hi , let u iy1 s u˜i be the character of Hiy1 relative to the monoid e iy1 Me iy1. Let

a G Ž u . s a G Ž V . s u 0 g char H0 .

Ž 14 .

If W is a C M-submodule of V, then clearly

a G Ž V . s a G Ž W . q a G Ž VrW . . PROPOSITION 1.1.

Ž 15 .

Let V be a C M-module. Consider a chain V s V0 = V1 = ??? = Vt s 0

of submodules of V such that RVi : Viq1 , i s 0, . . . , t y 1. Let Wi s VirViq1 and let x i denote the character of the CG-module j G Wi , i s 1, . . . , t. Then

x 1 q ??? qxt s

Ý Ž y1. l G a G Ž V . , Ž .

G

where the summation is o¨ er chains G of J-classes with G g G.

58

MOHAN S. PUTCHA

Proof. By Ž15. we are reduced to the case when t s 1. So let V be a semisimple C M-module. We prove the result by induction on < M <. Now Vs

[U j V . J

Jg

Let J g U , J / G. Then e J V is a semisimple C e J Me J-module. So by induction hypothesis the C H J-module e J j J V has character

Ý Ž y1. l V Ž

V

.

aV Ž eJ V . ,

where V ranges through chains of J-classes of e J Me J containing H J . For such a chain V, let VX denote the corresponding chain of J-classes of M and let G s VX j  G4 . Then < G < s < V < q 1 and the CG-module j J V has character y Ý Ž y1 . G

lŽ G .

aG Ž V . ,

where the summation is over chains G of J-classes of M such that G g G and J is the maximal element of G _  G4 . Since the CG-module V has character a G Ž V ., the CG-module j G V has the desired character. 2. HIGHEST WEIGHT CATEGORIES Motivated by the representation theory of Lie algebras and algebraic groups, Cline et al. w4x Žor see w11, Chap. 3x. introduced the concept of a highest weight category. They showed that the modules of a finite dimensional algebra A form a highest weight category if and only if A is quasi-hereditary. This means that 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 ? R ? Ik : Iky1 where R s rad A.

Such a chain of ideals is called a hereditary chain. We refer to w6x for the abstract properties of quasi-hereditary algebras. Let M be a finite regular monoid. Let the weight poset L be as in Ž10.. For u g L, let LŽ u . denote the corresponding simple C M-module. For u g Irr H J , u X g Irr H J X define u - u X if J X - J. For u g Irr H J , there is an

COMPLEX REPRESENTATIONS

59

associated primitive idempotent e of C M such that e F e J . Let the standard module, DŽ u . s C J ? e .

Ž 16 .

If e1 , . . . , e t are idempotent representatives of the L-classes of J, then clearly as C M modules, C J s C J ? e1 [ C J ? e2 [ ??? [ C J ? e t C J ? ei ( C J ? e J ,

i s 1, . . . , t.

Ž 17 .

By Ž16., Ž17., the standard modules are just the indecomposable components of the C M-modules C J, J g U . Now e ? C J is a right C M-module. So its dual =Ž u . s Ž e ? C J .

U

Ž 18 .

is a costandard Žleft. C M-module. THEOREM 2.1.

Let M be a finite regular monoid. Then

Ži. The C M-modules form a highest weight category. Žii. C M is a quasi-hereditary algebra with global dimension bounded by twice the length of a longest chain in U . Proof. Let A s C M, R s rad A. Ži. Let u g Irr H J . Correspondingly let e be a primitive idempotent of C M such that e F e J . Then LŽ u . ( AerR e. Let I s I X Ž J . as in Ž2., B s ArC I. Then D Ž e . s Be. So LŽ u . ( BerR e in B. By Ž8., J ? R e s 0 in B. If J X g U , J X h J, then clearly JX ? R e : JX ? C J s 0 in B. Hence for J X g U , J X ? R e / 0 in B « J X ) J. Thus the composition factors of R e in B have apex ) J. Thus if LŽ u X . is a composition factor of R e in B, then u X g Irr H J X , J X ) J. So u X - u . Clearly Ae is the projective cover of the simple module LŽ u . and AerC I ? e ( Be s D Ž u . .

60

MOHAN S. PUTCHA

We need to show that C I ? e has a filtration by some DŽ u X ., u X ) u . Let I0 s I and inductively Ikq 1 s  J X g U < J X - J Y for some J Y g U with J Y : Ik 4 . Then we have ideals I s I0 > I1 > I2 > ??? . So C I > C I1 > C I2 > ??? and C I ? e > C I1 ? e > C I2 ? e > ??? . Let Bk s ArC Ikq1 ,

k s 0, 1, . . . .

Let Ik _ Ikq1 s J1 j ??? j Jt , J1 , . . . , Jt g U . Then in Bk Ži.e., modulo C Ikq 1 ., C Ik s C J1 [ ??? [ C Jt . Also in Bk C Ik s C Ik ? e [ C Ik ? Ž 1 y e . . Hence in Bk , C Ik ? e is a direct sum of some indecomposable components of A-modules C Ji , i s 1, . . . , t. By Ž17. these are of the form DŽ u X ., u X ) u . Hence Žin A. for k s 0, 1, . . . , C Ik ? erC Ikq1 ? e is a direct sum of DŽ u X ., u X ) u. Žii. Let J g U , I s I X Ž J ., B s ArC I. Then in B, C J ? e J s Be J is a projective B-module. Hence by Ž17., C J is a projective B-module.

Ž 19 .

Let J1 denote the minimum J-class of M and let I1 s J1. Inductively let Ikq1 denote the union of Ik and all J-classes J g U such that J covers J X in U for some J X : Ik . Then I1 ; I2 ; ??? ; It s M. Clearly t y 1 is the length of a longest chain in U . Now 0 ; C I1 ; C I2 ; ??? ; C It s A

COMPLEX REPRESENTATIONS

61

is a hereditary chain by Ž8., Ž17., Ž19.. Hence by w4x Žor see w6x. A has global dimension F 2Ž t y 1.. By Theorem 2.1 and w4, Lemma 3.2Žb.x we have COROLLARY 2.2. Let u g Irr H J , u X g Irr H J X . Then Ext 1 Ž LŽ u ., LŽ u X .. / 0 implies that J ) J X or J X ) J. EXAMPLE 2.3. Let TnŽC. denote the algebra of n = n upper triangular matrices over C. Then TnŽC. is quasi-hereditary by w4, Example 3.3Ža.x. Let Ei j denote the usual elementary matrices, 1 F i F n, 1 F j F n. For i F j, let j

i

Fi j s

Ý ks1

Ek k q

Ý

Ei k .

ksiq1

Let M s  Fi j <1 F i F j F n4 . Then M is an idempotent monoid and TnŽC. ( C M. EXAMPLE 2.4. Let MnŽC. denote the algebra of n = n matrices over C. Let M s  0, I 4 j  Ei j <1 F i F n, 1 F j F n4 . Then M is an inverse monoid Žregular monoid with commuting idempotents. and C M ( MnŽC. [ C [ C. 3. MONOID QUIVERS Let M be a finite regular monoid. In this section we will introduce the concept of a monoid quiver and study its connections with the ordinary quiver of C M. We will then explicitly determine the monoid quiver of M in terms of some characters of the unit group G. The Žordinary. qui¨ er of C M is defined as follows, cf. w1, 11x. Let m u g Irr H J , u X g Irr H J X . Then define u ª u X if dim C Ext 1 Ž L Ž u . , L Ž u X . . s m ) 0. Then by Corollary 2.2, J ) J X or J X ) J. We write u ª u X if m s 1. We define the monoid qui¨ er of M as follows. Let u g Irr H J , u X g Irr H J X . Then u , u X are adjacent only if J - J X or J X - J. Let J - J X , I s I X Ž J . as in Ž2.. Let e F e J , e X F e J X , be primitive idempotents of C M

62

MOHAN S. PUTCHA m

corresponding respectively to u , u X . Define u ¸ u X if dim C e X R e s m ) 0

in C MrC I.

Ž 20 .

m

We write u ¸ u X if m s 1. Define u X ¸ u if dim C e R e X s m ) 0

in C MrC I.

Ž 21 .

We write u X ¸ u if m s 1. We note that for any J g U , the monoid quiver of e J Me J is obtained from the monoid quiver of M by restriction. LEMMA 3.1. Let J1 , J 2 g U , j i s j J i , i s 1, 2, as in Ž12.. Let I be an ideal of M such that for each J-class J of I, J - Ji , i s 1, 2,. Then j 1Ž R l C I . j 2 : R 2 where R s rad C M. Proof. We prove by induction on < I <. Let J be a maximal J-class of I, K s I _ J. Then K is an ideal of M. Let A s C M, R s rad A, B s ArC K. Then C J s C IrC K is an ideal of B. Let r g R l C I s rad C I. Then in B, r g rad C J. Let hJ be as in Ž11. and a s Ž 1 y hJ . r Ž 1 y hJ . g rad C I. So in B, a g rad C J. Let x g J. Then by Ž11., x Ž1 y hJ ., Ž1 y hJ . x g rad C J. So xa, ax g rad 2 C J. Now for some e, f g EŽ J ., x s ex s xf. So by Ž8., xa s ax s 0 in C J. So by Ž9., a g rad 2 C J. Hence in A, a g Ž R l C K . q R2. So by the induction hypothesis j 1 a j 2 g R 2 . Now by Ž12., j 1hJ , hJ j 2 g R. So

j 1 rj 2 g j 1 a j 2 q R 2 : R 2 . This completes the proof. THEOREM 3.2. Ži.

Let u g Irr H J , u X g Irr H J X . Then

m

n

If u ª u X then u ¸ u X with m F n. m1

n

m2

mt

Žii. If u ¸ u X , then u s u 0 ª u 1 ª ??? ª u t s u X for some u i g L, i s 1, . . . , t y 1. Žiii. m

m

m

If J X s G co¨ ers J in U , then u ª u X if and only if u ¸ u X , and m

u X ª u if and only if u X ¸ u . Živ. If rad 2 C M s 0, then the monoid qui¨ er is equal to the ordinary qui¨ er.

COMPLEX REPRESENTATIONS

63

Proof. Let A s C M, R s rad A. Let J - J X . The case J X - J follows by duality w1, II.3x. Let e F e J , e X F e J X be primitive idempotents of A m associated with u , u X respectively. Let I s I X Ž J . as in Ž2.. Let u ª u X . Then m s dim Ž e X R ere X R 2e . . Let B s ArC I. Let g denote the restriction to e X R e of the canonical map from A to B. Then g is linear. Let r g R such that g Ž e X re . s 0. Then e X re g R l C I s rad C I. By Lemma 3.1,

j J X ? e X re ? j J g R 2 . Since e F e J and e X F e J X are primitive idempotents, we see by Ž12., Ž13. that e ? j J y e , j J X ? e X y e X g R. Hence e X re g R 2 . Hence g is 1]1 and m s dim Ž e X R ere X R 2e .

in B.

Since n s dim e X R e in B by Ž20., m F n. This proves Ži.. If R 2 s 0, then m s n and Živ. is also proved. If J X s G covers J, then by Ž8., R 2e s 0 in B and hence again m s n. Thus Žiii. is also proved. n

Now assume u ¸ u X . Then n s dim Ž e X R e . ) 0

in B.

Let e X R te / 0, e X R tq1e s 0. Then there exist r 1 , . . . , rt g R such that e X rt ??? r 1 e / 0. So there exist primitive idempotents e 1 , . . . , e ty1 such that

e X rt e ty1 ??? r 2 e 1 r 1 e / 0. With e 0 s e , e t s e X , e i ri e iy1 f R 2 for i s 1, . . . , t. Let u i g L correspond to e i , i s 0, . . . , t. Then m1

m2

mt

u s u 0 ª u 1 ª ??? ª u t s u X for some positive integers m1 , . . . , m t . This proves Žii., completing the proof. We now wish to determine the monoid quiver explicitly. Let u g Irr H J . Then u˜ is the character of the CG-module LŽ u .. Let uq, uy denote the characters of the CG-modules DŽ u . and =Ž u ., respectively. uq and uy were determined by the author w18x as follows. For an R-class R of J, let P s P Ž R . s  g g G < gR s R 4 .

Ž 22 .

64

MOHAN S. PUTCHA

Then there is a natural homomorphism d : P ª H J such that U s U Ž R . s ker d s  g g G < gr s r for all r g R 4 .

Ž 23 .

Similarly for an L-class L of J, let Pys Py Ž L . s  g g G < Lg s L4 .

Ž 24 .

Then there is a natural homomorphism dy: Pyª H J such that Uys Uy Ž L . s ker dys  g g G < lg s l for all l g L4 .

Ž 25 .

R. Let R1 , . . . , R m denote the orbit representaG acts on the left on JrR tives. Let Pi s P Ž R i . and let d i : Pi ª H J denote the corresponding homomorphism, i s 1, . . . , m. So u g Irr H J lifts to w i g char Pi , i s L . Let L1 , . . . , L n denote the 1, . . . , m. Similarly G acts on the right on JrL y orbit representatives. Let Pjys Py Ž L j . and let dy j : Pj ª H J denote the corresponding homomorphism, j s 1, . . . , n. So u lifts to c j g char Pj , j s 1, . . . , n. It is shown by the author w18x that

uqs

m

Ý Ž wi ­G . ,

uys

is1

n

Ý Ž cj ­G . .

Ž 26 .

js1

Let J, J X g U , J X ) J, u g Irr H J . Let G s  J X s J 0 ) J1 ) ??? ) Jt 4 be a chain of J-classes with Jt ) J. By Ž1., we can assume that e t s e J t ) e J . Let Ht s H J t , q ˜ aq G Ž u . s aG Ž u y u . y ˜ ay G Ž u . s aG Ž u y u . ,

where a G is as in Ž14. and uqy u˜, uyy u˜g char Ht with respect to the monoid e t Me t . Let the virtual characters Žof H J X . X bq J Žu . s

Ý Ž y1. l G aqG Ž u .

X by J Žu . s

Ý Ž y1. l G ayG Ž u . ,

Ž .

G

Ž 27 .

Ž .

G

where the summations are over chains of J-classes G s  J X s J 0 ) J1 ) ??? ) Jt 4 with Jt ) J. THEOREM 3.3. Then

Let J, J X g U with J X ) J. Let u g Irr H J , u X g Irr H J X .

Ži.

. yX Ž . XŽ X bq J u , b J u are ordinary characters of H J .

Žii.

.. ) 0. XŽ u ¸ u X if and only if m s Ž u X , bq J u

Žiii.

.. ) 0. XŽ u X ¸ u if and only if m s Ž u X , by J u

m

m

COMPLEX REPRESENTATIONS

65

Proof. Without loss of generality let J X s G. Let A s C M, R s rad A. Let I s I X Ž J . as in Ž2., B s ArC I. Let e F e J be a primitive idempotent corresponding to u and let V s R e in B. Now by Ž16., DŽ u . s Be and hence V s R ? DŽ u .. The CG-module DŽ u . has character uq. So the CG-module V has character uqy u˜. Let R tq1 s 0, Vi s R i V. Then V s V0 = V1 = ??? = Vt s 0. Let Wi s VirViq1 , i s 0, . . . , t y 1. Then by Proposition 1.1, the CGmodule j G W0 [ j G W1 [ ??? [ j GWty1 qŽ . has character bG u . Let e X F j G be a primitive idempotent of A correX sponding to u . Then

e X W0 [ e X W1 [ ??? [ e X Wty1 X X .. XŽ has dimension Ž u X , bq J u . This is also clearly dim e V s dim e R e . By duality and Ž20., Ž21., the proof is complete.

By the proof of Proposition 1.1 we also have, COROLLARY 3.4.

Let J g U , u g Irr H J . Then &

Ži.

qŽ . X bG u s uqy u˜y Ý J - J X - G bq J Žu .

Žii.

yŽ . X bG u s uyy u˜y Ý J - J X - G by J Žu ..

&

For an idempotent monoid, the monoid quiver is particularly easy to compute. Let M be an idempotent monoid. Then U is a lattice and each J g U is a subsemigroup, cf. w3, Chap. 4x. Moreover each irreducible representation is of degree 1 and L ( U . Let J, J X g U , J - J X . Let nqŽ J, J X . denote the number of R-classes of e J X Je J X . Let nyŽ J, J X . denote the number of L-classes of e J X Je J X . Let bqŽ J, J X ., byŽ J, J X . be defined inductively by

nq Ž J , J X . s 1 q

Ý Y

J-J FJ X

ny Ž J , J . s 1 q

Ý Y

J-J FJ

X

X

bq Ž J , J Y . by Ž J , J Y . .

Ž 28 .

Let m denote the Mobius function on U , cf. w20, Chap. 3x. Then by the ¨ Mobius inversion formula ¨

bq Ž J , J X . s ym Ž J , J X . q

Ý Y

J-J FJ

by Ž J , J X . s ym Ž J , J X . q

Ý Y

J-J FJ

By Theorem 3.3 and Corollary 3.4 we have,

X

X

nq Ž J , J Y . m Ž J Y , J X . ny Ž J , J Y . m Ž J Y , J X . .

Ž 29 .

66

MOHAN S. PUTCHA

THEOREM 3.5.

Let M be an idempotent monoid, J, J X g U , J - J X . Then

Ži.

bqŽ J, J X . G 0, byŽ J, J X . G 0.

Žii.

J ¸ J X if and only if m s bq Ž J, J X . ) 0.

Žiii.

J X ¸ J if and only if m s by Ž J, J X . ) 0.

m

m

EXAMPLE 3.6. Consider the idempotent monoid M with multiplication table given by

1 a b c d e f g h i j

1

a

b

c

d

e

f

g

h

i

j

1 a b c d e f g h i j

a a a c d e f g h i j

b b b c d e f g h i j

c c c c g g g g h g g

d d d g d e f g g i g

e e e g e e e g g i g

f f f g f f f g g i g

g g g g g g g g g g g

h h h h h h h h h h h

i i i i i i i i i i i

j j j j j j j j j j j

Then U s  J1 , J 2 , J 3 , J4 , J5 , J6 4 , where J1 s  14 , J 2 s  a, b4 , J 3 s  c4 , J4 s  d4 , J5 s  e, f 4 , J6 s  g, h, i, j4 . The lattice U has the following structure: J1 J2 J3

J4 J5 J6

67

COMPLEX REPRESENTATIONS

By Theorems 3.3, 3.5, the monoid quiver which is also the ordinary quiver of C M is as follows:

6

6

J2

6

6

J1

J4

6

J3

J5

J6

Similarly we can construct idempotent monoids corresponding to all the Dynkin diagrams. Note that the path semigroup associated with a path algebra, cf. w1x, is never regular.

4. THE FULL TRANSFORMATION SEMIGROUP The full transformation semigroup Tn is the regular monoid of all self maps of n s  1, 2, . . . , n4 with respect to composition. It plays a pivotal role in semigroup theory and automata theory, cf. w3, 7x. Note that we are writing maps on the left while in w3, 2.2x they are written on the right. Tn has n n elements. If the Jk denote the set of maps of rank k, then U Ž Tn . s  J1 , . . . , Jn4 . Let e k g EŽ Jk ., with e1 - e2 - ??? - e n s 1. Then Sk s H Ž e k . is the symmetric group of degree k and e k Tn e k ( Tk , k s 1, . . . , n. In particular Sn s Jn is the unit group of Tn . By Ž10., L s L Ž Tn . s

"

1FmFn

Irr Sm .

Ž 30 .

Let l s Ž l1 , . . . , l k ., l1 G l2 G ??? G l k ) 0, Ý l i s n, be a partition of n. As usual w8x, we write l & n. Let l s Ž l 1 , . . . , l k . denote a corresponding partition of n, i.e., < l i < s l i , i s 1, . . . , k. Then the Young subgroup Sl s S l s S l 1 = ??? = S l k ( Sl1 = ??? = Sl k . Let w lx denote the associated irreducible character of Sn . In particular w n x, w1n x are the trivial and alternating character of S n , respectively. If g & k and l & n y k, then the outer product w g xw lx is obtained by inducing the outer tensor product character of Sk = Snyk ¨ Sn to S n . In particular w g xw n y k x is explicitly obtained by Young’s rule w8, Corollary 2.8.3x. The author w18, Theorem 2.1x showed that if w g x g Irr Sk , then in Tn , &

wg x s

½

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

if w g x / w 1k x if w g x s w 1k x .

Ž 31 .

68

MOHAN S. PUTCHA

For a : n, let R a denote the set of all f g Tn such that f Žn. s a. Then R a , a : n,
Ž 32 .

R. So by Ž26., Ž32., Sn acts transitively on the left on JkrR q

w g x s w g xw n y k x

if g & k.

Ž 33 .

Let l s Ž l1 , . . . , l k . & n and let l s Ž l 1 , . . . , l k . be a corresponding partition of n. Let L l denote the set of all f g Tn such that the non-empty fibres of f are exactly l 1 , . . . , l k . These are the L-classes of Jk . As in Ž25., let Ulys Uy Ž L l . s S l .

Ž 34 .

yŽ Let Py L l . be as in Ž24.. If s g Sn , then s g Py ls P l if and only if

s Ž l1. , . . . , s Ž l k . 4 s  l1, . . . , l k4 . Thus Py l s NS nŽ S l . ,

Ž 35 .

Py l rS l ¨ S k .

In fact Py l rS l is isomorphic to a Young subgroup of S k . S n acts on the L . The orbits are in 1]1 correspondence with partitions l of n right on JkrL y y with k parts. Write Ply, Uly for Py l , Ul , respectively. So Ul s Sl . If g & k, y then w g x lifts to w g x l g char Pl by Ž35.. So by Ž26., y

w g x s Ý w g x l ­ Sn ,

g & k,

Ž 36 .

l

where the summation is over l & n with k parts. By Ž34., Ž35., and Frobenius reciprocity, any component of w g x l ­ Sn is a component of w l1 x ??? w l k x .

Ž 37 .

If l1 , . . . , l k are distinct then Plys Sl . Hence if l1 , . . . , l k are distinct then w g x l ­ Sn s deg g ? w l1 x ??? w l k x . Ž 38 .

69

COMPLEX REPRESENTATIONS

If k s n y 1 then l s w2, 1ny 2 x, Plys S2 = Sny2 , Ulys S2 . Hence by Ž35., Ž36., y

w g x s Ž w g x x Sny 2 . w 2 x

if k s n y 1.

Ž 39 .

qw x yw x w x yw x Ž . For g & k, let bq n g , b n g denote bS n g , bS n g as in 27 . Then by Corollary 3.4, q

&

y

&

&

bq n wg x s wg x y wg x y

Ý

bq m wg x

k-m-n

Ž 40 .

&

by n wg x s wg x y wg x y

Ý

by m wg x .

k-m-n

We claim that k bq n w1 x s

½w

1n x 0

if k s n y 1 otherwise.

Ž 41 . &

We prove this by induction on n. If k s n y 1, then by Ž31., w1ny 1x s w2, 1ny 2 x and by Ž33., w1ny1 xq s w1ny1 xw1x s w2, 1ny2 x q w1n x. Hence w ny 1 x s w1n x. So let k - n y 1. Then by induction hypothesis, bq w kx bq n 1 m 1 s 0 if k q 1 - m - n y 1. Hence by Ž31., Ž33., Ž40., &

q

&

k k k bq y 1kq1 n w1 x s w1 x y 1

s w n y k q 1, 1ky 1 x q w n y k, 1k x y w n y k q 1, 1ky 1 x y w n y k, 1k x s 0. Also by Ž31., Ž33.,

bq n wg x s 0

if g & k , w g x / w 1k x .

Ž 42 .

By Ž34., Ž35., PŽyn. s UŽyn. s Sn . Hence by Ž40.,

by n w 1 x s 0.

Ž 43 .

Tn for small n. Since CT T1 ( C, we We now compute the quiver of CT T2 . By Ž41., Ž43., and Theorems 3.2, 3.3, the quiver is given by begin with CT T3 . By Ž39., Fig. 1. Next consider CT y

y

w 2 x s w 12 x s w 1 xw 2 x s w 3 x q w 2, 1 x .

70

MOHAN S. PUTCHA

6

w1x

w2x

w 12 x

FIGURE 1

Hence by Ž31., Ž40.,

by 3 w 2 x s 0,

2 by 3 w1 x s w3x .

Ž 44 .

Thus by Ž41., Ž43., Ž44., Theorems 3.2, 3.3, the monoid quiver equals the T4 . Then ordinary quiver and is given by Fig. 2. Next consider CT y y PŽ3, 1. s UŽ3 , 1. s S3 = S1 y UŽ2, 2. s S2 = S2 .

If w g x g Irr S2 , then by Ž38.

w g x Ž3, 1. ­ S4 s w 3 xw 1 x s w 4 x q w 3, 1 x . Now

y PŽ2, 2.

y rUŽ2, 2. (

Ž 45 .

S2 and by Ž37.,

w 2 x Ž2 , 2. ­ S4 q w 12 x Ž2, 2. ­ S4 s w 2 xw 2 x s w 4 x q w 3, 1 x q w 2 2 x . Clearly w2xŽ2, 2. ­ S4 has degree 3 and has w4x as a component. Hence

w 2 x Ž2, 2. ­ S4 s w 4 x q w 2 2 x w 12 x Ž2, 2. ­ S4 s w 3, 1 x . So by Ž36., Ž45., Ž46., y

w 2 x s 2 w 4 x q w 3, 1 x q w 2 2 x y

w 12 x s w 4 x q 2 w 3, 1 x . By Ž31.,

w˜2 x s w 2 xw 2 x s w 4 x q w 3, 1 x q w 2 2 x &

12 s w 3, 1 x . w 12 x

6

6

w1x

6

w2x

w3x

w 2, 1 x

FIGURE 2

w 13 x

Ž 46 .

COMPLEX REPRESENTATIONS

71

Hence by Ž31., Ž40., Ž44.,

by 4 w2x s w4x 2 ˜ by 4 w 1 x s w 4 x q w 3, 1 x y w 3 x s 0.

Clearly,

Ž 47 .

w 3 x x S2 s w 2 x w 2, 1 x x S2 s w 2 x q w 12 x w 13 x x S2 s w 12 x .

Hence by Ž39., y

w 3 x s w 2 xw 2 x s w 4 x q w3, 1 x q w2 2 x y

w 2, 1 x s w 2 xw 2 x q w 12 xw 2x s w 4 x q 2 w 3, 1x q w 2, 12 x q w 2 2 x y

w 13 x s w 12 xw 2 x s w 3, 1 x q w 2, 12 x . By Ž31.,

w˜3 x s w 4 x q w 3, 1 x

&

2, 1 s w 3, 1 x q w 2 2 x q w 2, 12 x &

13 s w 2, 12 x . Hence by Ž40., 2 by 4 w3x s w2 x

by 4 w 2, 1 x s w 4 x q w 3, 1 x

Ž 48 .

3 by 4 w 1 x s w 3, 1 x . By Ž41., Ž43., Ž44., Ž47., Ž48., Theorems 3.2, 3.3, the monoid quiver equals the ordinary quiver and is given by Fig. 3. T2 , CT T3 are of finite representation Ponizovskii w15x has shown that CT Tn is of finite representation type for all n. type and has conjectured that CT We now answer this conjecture in the negative.

THEOREM 4.1. Ži. CT T2 , CT T3 are of finite representation type. Žii. CT T4rR 2 is of finite representation type. Žiii. For n G 5, CT TnrR 2 and hence CT Tn is not of finite representation type. Proof. Ži. This follows from w1, Theorems III.1.9, VIII.5.4x since the quivers Žsee Figs. 1, 2. are unions of Dynkin diagrams. Žii. This follows from w1, Theorem X.2.6x since the separated quiver of the quiver in Fig. 3 is a union of Dynkin diagrams.

72

MOHAN S. PUTCHA

w 13 x

6

w3x

6

w12 x

6

6

w1x

6

w14 x

w 3, 1 x

6

6

w2 x

w 2, 1 x

2

w 2, 12 x

6

w4x 6

w2x FIGURE 3

Žiii. First consider T5 . By w8, Theorem 2.4.3x,

w 3, 1 x x S3 s w 3 x q w 2, 1 x . So by Ž39., y

w 3, 1 x s w 3 xw 2 x q w 2, 1 xw 2 x s w 5 x q w 4, 1 x q w 3, 2 x q w 4, 1 x q w 3, 2 x q w 3, 12 x q w 2 2 , 1 x s w 5 x q 2 w 4, 1 x q 2 w 3, 2 x q w 3, 12 x q w 2 2 , 1 x . By Ž31., &

3, 1 s w 3, 1 xw 1 x s w 4, 1 x q w 3, 2 x q w 3, 12 x . Hence by Ž40., 2 by 5 w 3, 1 x s w 5 x q w 4, 1 x q w 3, 2 x q w 2 , 1 x .

By Theorems 3.2, 3.3, we have

w 5 x ª w 3, 1 x w 4, 1 x ª w 3, 1 x w 3, 2 x ª w 3, 1 x w 2 2 , 1 x ª w 3, 1 x .

COMPLEX REPRESENTATIONS

73

T5 is not a union of Dynkin It follows that the separated quiver of CT T5rR 2 is not of finite representation diagrams. By w1, Theorem X.2.6x, CT type. Now let n ) 5. By w8, Theorem 2.4.3x,

w n y 2, 1 x x Sny 2 s w n y 2 x q w n y 3, 1 x . So by Ž39., y

w n y 2, 1 x s w n y 2 xw 2 x q w n y 3, 1 xw 2 x s w n x q w n y 1, 1 x q w n y 2, 2 x q w n y 1, 1 x q w n y 2, 2 x q w n y 2, 12 x q w n y 3, 3 x q w n y 3, 2, 1 x s w n x q 2 w n y 1, 1 x q 2 w n y 2, 2 x q w n y 2, 12 x q w n y 3, 3 x q w n y 3, 2, 1 x . By Ž31., &

n y 2, 1 s w n y 2, 1 xw 1 x s w n y 1, 1 x q w n y 2, 2 x q w n y 2, 12 x . Hence by Ž40.,

by n w n y 2, 1 x s w n x q w n y 1, 1 x q w n y 2, 2 x q w n y 3, 3 x q w n y 3, 2, 1 x . By Theorems 3.2, 3.3,

w n x ª w n y 2, 1 x w n y 1, 1 x ª w n y 2, 1 x w n y 2, 2 x ª w n y 2, 1 x w n y 3, 3 x ª w n y 2, 1 x w n y 3, 2, 1 x ª w n y 2, 1 x . Tn is not a union of Dynkin It follows that the separated quiver of CT TnrR 2 is not of finite representation diagrams. By w1, Theorem X.2.6x, CT type. This completes the proof. Tn has no multiple edges. Conjecture 4.2. The quiver of CT Tn . It is well known w10, Theorem We will now determine the blocks of CT 22.9x that the blocks of an algebra A are in 1]1 correspondence with those

74

MOHAN S. PUTCHA

of Arrad 2 A and hence with the components of the ordinary quiver of A. So for a finite regular monoid M, we see by Theorem 3.2 that the blocks of C M are in 1]1 correspondence with the components of the monoid quiver of M. T2 has two blocks and CT T3 has three blocks. By Figs. 1 and 2, CT THEOREM 4.3.

Let n G 4. Then

Ži. CT Tn has two blocks with the block of w2, 1ny 2 x being isolated. Žii. The block of w2, 1ny 2 x is a simple algebra of dimension Ž n y 1. 2 . TnrK is Proof. If Ži. is true then Tn _ Sn is contained in a block K. So CT a homomorphic image of the semisimple algebra CSn and hence Žii. is valid. So it suffices to prove Ži.. We do this by induction on n. For n s 4, this is clear from Fig. 3. So let n ) 4. Write ; for the equivalence relation on L of being in the same block. Let X denote the block of w1x in L. By the induction hypothesis w g x g X if g & m - n, g / Ž2, 1ny 3 .. Let l s Ž l1 , . . . , l k . & n, l / Ž2, 1ny 2 .. If l1 s 1, then w lx s w1n x ; w1x g X by Theorem 3.3, Ž41.. So let l1 G 2. Suppose l k G 2. Let g s Ž l1 q 1, l2 , . . . l ky1 , l k y 2. & n y 1. By Ž31., w lx is not a component of w g ˜ x. Let g X s Ž l1 , l2 , . . . , l k y 2. & n y 2. Then by w8, Theorem 2.4.3x, w g X x is a component of w g xx Sny 2 . Clearly w lx is a component of w g X xw2x. So by Ž39., w lx is a component of w g xy. By Ž40., w lx is a component of by w x n g . By Theorem 3.3, w lx ; w g x g X. Next let l k s 1. Let i be maximal such that l i ) 1. Suppose i ) 1. Let g s Ž& l1 q 1, l2 , . . . , l i y 1, 1ky iy1 . & n y 1. Then w lx is not a component of w g x by Ž31.. Let g X s Ž l1 , l 2 , . . . , l i y 1, 1ky iy1 . & n y 2. Then w g X x is a component of w g xx Sny 2 and w lx is a component on w g X xw2x. By Ž39., w lx is a component of w g xy. By Ž40., Theorem 3.3, w lx ; w g x g X. We are therefore left with considering the case when l s Ž m, 1ny m ., m G 3. Let g s Ž m y 1, 2, 1ny my2 . & n y 1. & Then w lx is not a component of w g x by Ž31.. Let g X s Ž m y 1, 1nymy1 . & n y 2. Then w g X x is a component of w g xx Sny 2 and w lx is a component of w g X xw2x. By Ž39., w lx is a component of w g xy. By Ž40., Theorem 3.3, w lx ; w g x g X. Now

w 2, 1ny 3 x x Sny 2 s w 2, 1ny 4 x q w 1ny 3 x .

Ž 49 .

By Ž39., y

w 2, 1ny3 x s w 2, 1ny4 xw 2 x q w 1ny2 xw 2 x .

Ž 50 .

Thus w4, 1ny 4 x is a component of w2, 1ny 3 xy. By Ž31., Ž40., Theorem 3.3, w2, 1ny 3 x ; w4, 1ny 4 x g X. Hence we have shown that X j w2, 1ny 2 x4 s L. We now show that w2, 1ny 2 x f X. Otherwise by Ž42., w2, 1ny 2 x is a w x w x Ž . w ny 2 x is a component of by n g for some g g Irr S k , k - n. By 40 , 2, 1

COMPLEX REPRESENTATIONS

75

component of w g xy. By Ž36., Ž37., w2, 1ny 2 x is a component w l1 xw l2 x ??? w l k x for some partition l s Ž l1 , . . . , l k . of n. This implies by w8, Theorem 2.8.5x that k s n y 1. So by Ž39., w2, 1ny 2 x is a component of w g X xw2x for some component w g X x of w g xx Sny 2 . Then w g X x s w1ny 2 x. So w g x s w2, 1ny 3 x or w1ny 1 x. First let w g x s w2, 1ny 3 x. Then by Ž50., w2, 1ny 2 x is a component of w g xy with multiplicity one. By Ž31., it is also a component of w g ˜ x. Hence w2, 1ny 2 x is not a component of by w x n g , a contradiction. Finally assume that w g x s w1ny 1 x. Then by Ž39., y

w g x s Ž w g x x Sny 2 . w 2 x s w 1ny 2 xw 2 x s w 3, 1ny 3 x q w 2, 1ny 2 x . w x By Ž31., w g ˜ x s w2, 1ny 2 x. Hence w2, 1ny 2 x is not a component by n g , a contradiction. Thus w2, 1ny 2 x is a block by itself, completing the proof.

REFERENCES 1. M. Auslander, I. Reiten, and S. Smalo, ‘‘Representation Theory of Artin Algebras,’’ Cambridge Studies in Advanced Mathematics, Vol. 36, Cambridge Univ. Press, Cambridge, UK, 1995. 2. A. H. Clifford, Matrix representations of completely simple semigroups, Amer. J. Math. 64 Ž1942., 327]342. 3. A. H. Clifford and G. B. Preston, ‘‘Algebraic Theory of Semigroups,’’ Vol. 1, Amer. Math. Soc. Surveys, Vol. 7, Amer. Math. Soc., Providence, 1961. 4. E. Cline, B. Parshall, and L. Scott, Finite dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 Ž1988., 85]99. 5. C. W. Curtis and I. Reiner, ‘‘Methods of Representation Theory with Applications to Finite Groups and Orders,’’ Vol. 1, Wiley, New York, 1981. 6. V. Dlab and C. M. Ringel, Quasi-hereditary algebras, Illinois J. Math. 33 Ž1989., 280]291. 7. J. Fountain ŽEd.., ‘‘Semigroups, Formal Languages and Groups,’’ NATO ASI Series, C 466, Kluwer Academic, Dordrecht, 1995. 8. G. James and A. Kerber, ‘‘The Representation Theory of the Symmetric Group,’’ Encyclopedia of Mathematics and Its Applications, Vol. 16, Addison]Wesley, Reading, MA, 1981. 9. G. Lallement and M. Petrich, Irreducible matrix representations of finite semigroups, Trans. Amer. Math. Soc. 139 Ž1969., 393]412. 10. T. Y. Lam, ‘‘A First Course in Noncommutative Rings,’’ Springer-Verlag, New YorkrBerlin, 1991. 11. S. Martin, ‘‘Schur Algebras and Representation Theory,’’ Cambridge Tracts, Vol. 112, Cambridge Univ. Press, Cambridge, UK, 1993. 12. W. D. Munn, Irreducible matrix representations of semigroups, Quart. J. Math. Oxford Ser. Ž 2 . 11 Ž1960., 295]309. 13. J. Okninski, ‘‘Semigroup Algebras,’’ Dekker, New York, 1991. ´

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14. J. Okninski ´ and M. S. Putcha, Parabolic subgroups and cuspidal representations of finite monoids, Internat. J. Alg. Comp. 1 Ž1991., 33]47. 15. J. S. Ponizovskii, Some examples of finite representations type semigroup algebras, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklo¨ . Ž LOMI . 160 Ž1987., 229]238. 16. M. S. Putcha, Sandwich matrices, Solomon algebras and Kazhdan]Lusztig polynomials, Trans. Amer. Math. Soc. 340 Ž1993., 415]428. 17. M. S. Putcha, Hecke algebras and monoid conjugacy classes, J. Algebra 173 Ž1995., 499]517. 18. M. S. Putcha, Complex representations of finite monoids, Proc. London Math. Soc. 73 Ž1996., 623]641. 19. M. S. Putcha and L. E. Renner, The canonical compactification of a finite group of Lie type, Trans. Amer. Math. Soc. 337 Ž1993., 305]319. 20. R. Stanley, ‘‘Enumerative Combinatorics,’’ Vol. 1, Wadsworth, Belmont, CA, 1986.