q-rook monoid algebras, Hecke algebras, and Schur-Weyl duality

Transcription

1 q-rook monoid algebras, Hecke algebras, and Schur-Weyl duality Tom Halverson Department of Mathematics and Computer Science Macalester College St Paul, MN and Arun Ram Department of Mathematics University of Wisconsin, Madison Madison, WI In memory of Sergei Kerov When we were at the beginnings of our careers Sergei s support helped us to believe in our work He generously encouraged us to publish our results on Brauer and Birman-Murakami-Wenzl algebras, results which had in part, or possibly in total, been obtained earlier by Sergei himself He remains a great inspiration for us, both mathematically and in our memory of his kindness, modesty, generosity, and encouragement to the younger generation 0 Introduction The rook monoid R k is the monoid of k k matrices with entries from {0, 1} and at most one nonzero entry in each row and column Recently, the representation theory of its Iwahori- Hecke algebra R k (q), called the q-rook monoid algebra, has been analyzed In particular, a Schur-Weyl type duality on tensor space was found for the q-rook monoid algebra and its irreducible representations were given explicit combinatorial constructions In this paper we show that, in fact, the q-rook monoid algebra is a quotient of the affine Hecke algebra of type A With this knowledge in hand, we show that the recent results on the q-rook monoid algebras actually come from known results about the affine Hecke algebra In particular (a) The recent combinatorial construction of the irreducible representations of R k (q) by Halverson [Ha] turns out to be a special case of the construction of irreducible calibrated representations of affine Hecke algebras of Cherednik [Ch] (see also Ram [Ra]), the construction of irreducible representations of cyclotomic Hecke algebras by Ariki and Koike Research supported in part by National Science Foundation grant DMS Research supported in part by National Science Foundation grant DMS , the National Security Agency and EPSRC grant GR K99015

2 2 tom halverson and arun ram [AK], and the construction of the irreducible representations of Iwahori-Hecke algebras of type B by Hoefsmit [Ho] (b) The Schur-Weyl duality for the q-rook monoid algebra discovered by Solomon [So2-4] and studied by Halverson [Ha] turns out to be a special case of the Schur-Weyl duality for cyclotomic Hecke algebras given by Sakamoto and Shoji [SS] Though these results show that the representation theory of the q-rook monoid algebra is just a piece of the representation theory of the affine Hecke algebra, this was not at all obvious at the outset It was only on the analysis of the recent results in [So4] and [Ha] that the similarity to affine Hecke algebra theory was noticed This observation then led us to search for and establish a concrete connection between these algebras The q-rook monoid algebra was first studied in its q = 1 version in the 1950 s by Munn [Mu1-2] Solomon [So1] discovered the general q-version of the algebra as a Hecke algebra (double coset algebra) for the finite algebraic monoid M n (F q )ofn n matrices over a finite field with q elements, with respect to the Borel subgroup B of invertible upper triangular matrices Later Solomon [So2] found a Schur-Weyl duality for R k (1) in which R k (1) acts as the centralizer algebra for the action of the general linear group GL n (C) onv k where V = L(ε 1 ) L(0) is the direct sum of the fundamental n-dimensional representation and the trivial module L(0) for GL n (C) Then Solomon [So3,4] gave a presentation of R k (q) by generators and relations and defined an action of R k (q) on tensor space Halverson [Ha] found a new presentation of R k (q) and used it to show that Solomon s action of R k (q) on tensor space extends the Schur-Weyl duality so that R k (q) is the centralizer of the quantum general linear group U q gl(n) onv k where now V = L(ε 1 ) L(0) is the direct sum of the fundamental and the trivial module for U q gl(n) Halverson also exploited his new presentation to construct, combinatorially, all the irreducible representations of R k (q) when R k (q) is semisimple The main results of this paper are the following: (a) We find yet another presentation (16) of R k (q) by generators and relations (b) Our new presentation shows that R k (q) =H k (0, 1; q)/i, where H k (0, 1; q) is the Iwahori-Hecke algebra of type B k with parameters specialized to 0 and 1, and I is the ideal generated by the minimal ideal of H 2 (0, 1; q) corresponding to the pair of partitions λ = ((1 2 ), ) (c) We show that the irreducible representations of R k (q) found in [Ha] come from the constructions of irreducible representations of H k (0, 1; q) (d) We use the fact that R k (q) is a quotient of H k (0, 1; q) and the fact that the Iwahori-Hecke algebra H k (q) oftypea k 1 is a quotient of R k (q) to easily determine, in Corollary 221, the values of q for which R k (q) is semisimple These values were first found in [So4] using other methods (e) We show that the Schur-Weyl duality between R k (q) and U q gl(n) comes from the Schur- Weyl duality of Sakamoto and Shoji [SS] for the cyclotomic Hecke algebras (Theorem 35) (f) We give a different Schur-Weyl duality for algebras A k (u 1,u 2 ; q) =H k (u 1,u 2 ; q)/i, where u 1,u 2 0, H k (u 1,u 2 ; q) is the Iwahori-Hecke algebra of type B k and I is the ideal generated by the minimal ideal of H 2 (u 1,u 2 ; q) corresponding to the pair of partitions λ = ((1 2 ), ) This Schur-Weyl duality comes from the Schur-Weyl duality of Orellana and Ram for the affine Hecke algebra (Theorem 33)

3 rook monoid algebras and hecke algebras 3 Acknowledgements A Ram thanks the Isaac Newton Institute for the Mathematical Sciences at Cambridge University for support for a very pleasant residency during which the research in this paper was completed 1 Presentations of the q-rook monoid algebras Fix q C The q-rook monoid algebra is the algebra R k (q) given by generators with relations P 1,P 2,,P k and T 1,T 2,,T k 1 (A1) Ti 2 =(q q 1 )T i +1, 1 i k 1, (A2) T i T i+1 T i = T i+1 T i T i+1, 1 i k 2, (A3) T i T j = T j T i, i j > 1, (R1) Pi 2 = P i, 1 i k, (R2) P i P j = P j P i, 1 i, j k, (R3) P i T j = T j P i, 1 i<j k, (R4) P i T j = T j P i = qp i, 1 j<i k, (R5) P i+1 = qp i T 1 i P i = q(p i T i P i (q q 1 )P i ), 1 i k 1 (11) The algebra R k (q) was introduced by Solomon [So1] as an analogue of the Iwahori-Hecke algebra for the finite algebraic monoid M k (F q )ofk k matrices over a finite field with q elements with respect to its Borel subgroup of invertible upper triangular matrices The presentation of R k (q) given above is due to Halverson [Ha] When q =1,R k (q) specializes to the algebra of the rook monoid R k that consists of k k matrices with entries from {0, 1} and at most one nonzero entry in each row and column These correspond with the possible placements of nonattacking rooks on an k k chessboard In this specialization, T i becomes the matrix obtained by switching rows i and i + 1 in the identity matrix I and, for 1 i k 1, P i becomes the matrix E i+1,i+1 + E i+1,i E k,k, where E i,j is the matrix with a1inposition (i, j) and zeros elsewhere The generator P k specializes to the 0 matrix (which is not the 0 element in the monoid algebra) Remark 12 The definition of R k (q) in [So1,3,4] and [Ha] uses generators T i in place of T i These generators satisfy T i 2 =(q 1) T i + q in place of (A1) In our presentation, if we let T i = qt i, then T i 2 = q 2 ((q q 1 )T i +1)=(q 2 1)qT i + q 2 =(q 2 1) T i + q 2, which shows that our algebra is the same except with parameter q 2 instead of q Define so that X i+1 = T i X i T i X i = T i 1 T i 2 T 1 (1 P 1 )T 1 T 2 T i 1, 1 i k, (13) Lemma 14 In R k (q) we have the following relations (a) P i P j = P j P i = P j, for i j (b) P 1 X 2 = P 1 P 2 Proof (a) If i = j this is (R1) If i<jthen, by (R5) and induction, P i P j = P i (P j 1 T j 1 P j 1 (q q 1 )P j 1 )=P j 1 T j 1 P j 1 (q q 1 )P j 1 = P j

4 4 tom halverson and arun ram (b) We use relations (A1), (R4), and (R5) to get P 1 X 2 = P 1 T 1 (1 P 1 )T 1 = P 1 T1 2 P 1 T 1 P 1 T 1 =(q q 1 )P 1 T 1 + P 1 P 1 T 1 P 1 T 1 =(q q 1 )P 1 T 1 + P 1 q 1 P 2 T 1 (q q 1 )P 1 T 1 = P 1 q 1 P 2 T 1 = P 1 P 2 Proposition 15 The q-rook monoid algebra R k (q) is generated by the elements X 1,T 1,,T k 1, and these elements satisfy the relations (A1), (A2), (A3), and (B1) X 1 T j = T j X 1, for 2 j k, (B2) X1 2 = X 1, (B3) X 1 T 1 X 1 T 1 = T 1 X 1 T 1 X 1, (B4) (1 X 1 )(T 1 q)(1 X 1 )(1 X 2 )=0, where X 2 = T 1 X 1 T 1 Proof By (R5), R n (q) is generated by X 1 =1 P 1,T 1,,T k 1 (B1) By (R3), X 1 T j =(1 P 1 )T j = T j (1 P 1 )=T j X 1 (B2) By (R1), X1 2 =(1 P 1 ) 2 =1 2P 1 + P1 2 =1 P 1 = X 1 (B3) Using (R4) and (R5), q(t 1 P 1 T 1 P 1 (q q 1 )T 1 P 1 )=T 1 P 2 = P 2 T 1 = q(p 1 T 1 P 1 T 1 (q q 1 )P 1 T 1 ), and so P 1 T 1 P 1 T 1 = T 1 P 1 T 1 P 1 +(q q 1 )(P 1 T 1 T 1 P 1 ) ( ) Now, using ( ) and (A1), X 1 T 1 X 1 T 1 =(1 P 1 )T 1 (1 P 1 )T 1 = T 2 1 P 1 T 2 1 T 1 P 1 T 1 + P 1 T 1 P 1 T 1 = T1 2 (q q 1 )P 1 T 1 P 1 T 1 P 1 T 1 + T 1 P 1 T 1 P 1 +(q q 1 )(P 1 T 1 T 1 P 1 ) = T1 2 P 1 T 1 P 1 T 1 + T 1 P 1 T 1 P 1 (q q 1 )T 1 P 1 = T1 2 T1 2 P 1 T 1 P 1 T 1 + T 1 P 1 T 1 P 1 = T 1 (1 P 1 )T 1 (1 P 1 ) = T 1 X 1 T 1 X 1 Finally, to show (B4), we use Lemma 14(b), (1 X 1 )(T 1 q)(1 X 1 )(1 X 2 )=P 1 (T 1 q)p 1 (1 X 2 ) = P 1 T 1 P 1 qp 1 P 1 T 1 P 1 X 2 + qp 1 X 2 = P 1 T 1 P 1 qp 1 P 1 T 1 (P 1 P 2 )+q(p 1 P 2 ) = qp 2 + P 1 T 1 P 2 = qp 2 + qp 1 P 2 by (R4) = qp 2 + qp 2 by Lemma 14(a) =0

5 rook monoid algebras and hecke algebras 5 Define a new algebra A k (q) by generators and relations X 1 and T 1,T 2,,T k 1 (A1) Ti 2 =(q q 1 )T i +1, 1 i k 1, (A2) T i T i+1 T i = T i+1 T i T i+1, 1 i k 2, (A3) T i T j = T j T i, i j > 1, (B1) X 1 T j = T j X 1, 2 j k, (B2) X1 2 = X 1, (B3) X 1 T 1 X 1 T 1 = T 1 X 1 T 1 X 1, (B4) (1 X 1 )(T 1 q)(1 X 1 )(1 X 2 )=0, where X 2 = T 1 X 1 T 1 (16) We will show that R k (q) = A k (q) Define P 1 =(1 X 1 ) and P i+1 = q(p i T i P i (q q 1 )P i ), 1 i k 1 (17) Lemma 18 Relations (B1)-(B4) are equivalent, respectively, to (B1 ) P 1 T j = T j P 1, for 2 j n, (B2 ) P1 2 = P 1, (B3 ) P 2 T 1 = T 1 P 2, (B4 ) P2 2 = P 2 Proof Subtracting T j from each side of T j P 1 T j =(1 P 1 )T j = X 1 T j = T j X 1 = T j (1 P 1 )=T j T j P 1, shows that (B1) is equivalent to (B1 ) Relations (B2) and (B2 ) are equivalent since 1 P 1 = X 1 = X1 2 =(1 P 1 ) 2 =1 2P 1 + P1 2 Since X 1 T 1 X 1 T 1 =(1 P 1 )T 1 (1 P 1 )T 1 =(1 P 1 )T1 2 T 1 P 1 T 1 + P 1 T 1 P 1 T 1 =(1 P 1 )((q q 1 )T 1 +1) T 1 P 1 T 1 +(q 1 P 2 +(q q 1 )P 1 )T 1 (by (A1) and (17)) =(q q 1 )T 1 +1 (q q 1 )P 1 T 1 P 1 T 1 P 1 T 1 + q 1 P 2 T 1 +(q q 1 )P 1 T 1 =(q q 1 )T 1 +1 P 1 T 1 P 1 T 1 + q 1 P 2 T 1 is equal to T 1 X 1 T 1 X 1 = T 1 (1 P 1 )T 1 (1 P 1 )=T1 2 (1 P 1 ) T 1 P 1 T 1 + T 1 P 1 T 1 P 1 =((q q 1 )T 1 + 1)(1 P 1 ) T 1 P 1 T 1 + T 1 (q 1 P 2 +(q q 1 )P 1 ) (by (A1) and (17)) =(q q 1 )T 1 +1 (q q 1 )T 1 P 1 P 1 T 1 P 1 T 1 + q 1 T 1 P 2 +(q q 1 )T 1 P 1 =(q q 1 )T 1 +1 P 1 T 1 P 1 T 1 + q 1 T 1 P 2, (B3) is equivalent to (B3 )

6 6 tom halverson and arun ram Expanding (T 1 q)(1 X 2 )=(T 1 q)(1 T 1 X 1 T 1 )=(T 1 q)(1 T 1 (1 P 1 )T 1 ) =(T 1 q)(1 (q q 1 )T 1 1+T 1 P 1 T 1 ) (by (A1)) =(T 1 q)t 1 ( (q q 1 )+P 1 T 1 ) =((q q 1 )T 1 +1) qt 1 )(q 1 q + P 1 T 1 ) (by (A1)) =(1 q 1 T 1 )(q 1 q + P 1 T 1 ) = q 1 q + P 1 T 1 q 2 T 1 + T 1 q 1 T 1 P 1 T 1 = (q q 1 )+q 1 (q q 1 )T 1 + P 1 T 1 q 1 T 1 P 1 T 1, gives (1 X 1 )(T 1 q)(1 X 2 )(1 X 1 )=P 1 (T 1 q)(1 X 2 )P 1 = (q q 1 )P 1 + q 1 (q q 1 )P 1 T 1 P 1 + P 1 T 1 P 1 q 1 P 1 T 1 P 1 T 1 P 1 (by (B2 )) = (q q 1 )P 1 +(2 q 2 )P 1 T 1 P 1 q 1 (P 1 T 1 P 1 ) 2 (by (B2 )) = (q q 1 )P 1 +(2 q 2 )(q 1 P 2 +(q q 1 )P 1 ) q 1 (q 1 P 2 +(q q 1 )P 1 ) 2 (by (17)) =(q q 1 )( 1+2 q 2 )P 1 + q 1 (2 q 2 )P 2 q 1 (q 2 P2 2 +2q 1 (q q 1 )P 2 +(q q 1 ) 2 P 1 ) =(q q 1 ) 2 (q 1 q 1 )P 1 + q 1 (q 2 P 2 q 2 P2 2 ) = q 3 (P 2 P2 2 ) and so (1 X 1 )(T 1 q)(1 X 2 )(1 X 1 ) = 0 if and only if P 2 2 = P 2 Thus (B4) is equivalent to (B4 ) Proposition 19 The algebra A k (q) is generated by T 1,,T k 1,P 1,,P k, and these elements satisfy the relations (A1), (A2), (A3), and (E1) P i T j = T j P i, for 1 i<j k, (E2) P i P j = P j P i = P i, for all 1 j<i k, (E3) Pi 2 = P i, for 1 i k, (E4) P i T j = T j P i = qp i, for 1 j<i k, (E5) P i+1 = qp i T 1 i P i = q(p i T i P i (q q 1 )P i ), for 1 i k 1 Proof Since X 1 =1 P 1, the elements T 1,,T n 1,P 1 generate A k (q) Relation (E5) is the definition of P i+1 We prove (E1) by induction on i The case i = 1 is (B1 ) Assume that j>i+1> 1, then T j commutes with P i by induction and T j commutes with T i by (A3), so T j P i+1 = q 1 T j P i T j P i (q q 1 )T j P i = q 1 P i T j P i T j (q q 1 )P i T j = P i+1 T j, proving (E1) We now prove (E2)-(E4) collectively by induction on i The case i = 1 for (E2) follows from (B2 ) since P 2 P 1 = q(p 1 T 1 P 1 (q q 1 )P 1 )P 1 = q(p 1 T 1 P 1 (q q 1 )P 1 )=P 2 ( )

7 rook monoid algebras and hecke algebras 7 The relation P 1 P 2 = P 2 is similar The first two cases of (E3) are (B2 ) and (B4 ) The i = 1 case of (E4) follows from (B4 ) since P2 2 = P 2 q(p 1 T 1 P 1 (q q 1 )P 1 ) = q(p 2 T 1 P 1 (q q 1 )P 2 ) (by ( )) = q(t 1 P 2 P 1 (q q 1 )P 2 ) (by (B3 )) = q(t 1 P 2 (q q 1 )P 2 ) (by ( )) Since P2 2 = P 2,wehaveT 1 P 2 = qp 2 The case P 2 T 1 = qp 2 is similar Now fix i>1 and assume the following relations, (E2 ) P i P j = P j P i = P i, for all 1 j<i, (E3 ) Pi 2 = P i, (E4 ) P i T j = T j P i = qp i, for 1 j<i, We show each of these relations for i +1 For (E2) we use (E3 )toget P i+1 P i = q(p i T i P 2 i (q q 1 )P 2 i )=q(p i T i P i (q q 1 )P i )=P i+1, and when j<i, we use (E2 )toget P i+1 P j = q(p i T i P i P j (q q 1 )P i P j )=q(p i T i P i (q q 1 )P i )=P i+1 The relations P i P i+1 = P i+1 and P j P i+1 = P i+1 are similar, and so (E2) is established To establish (E3), let i>2 (note that we have established i =1, 2), P 2 i+1 = q 2 (P i T i P i (q q 1 )P i ) 2 = q 2 (P i T i P i T i P i 2(q q 1 )P i T i P i +(q q 1 ) 2 P i ) (by (E3 )) = q 2 (P i T i qp i 1 T i 1 P i 1 T i P i q(q q 1 )P i T i P i 1 T i P i 2(q q 1 )P i T i P i +(q q 1 ) 2 P i ) (by (E5)) = q 2 (qp i P i 1 T i T i 1 T i P i 1 P i q(q q 1 )P i P i 1 Ti 2 P i 2(q q 1 )P i T i P i +(q q 1 ) 2 P i ) (by (E1)) = q 2 (qp i T i T i 1 T i P i q(q q 1 )P i Ti 2 P i 2(q q 1 )P i T i P i +(q q 1 ) 2 P i ) (by (E2)) = q 2 (qp i T i 1 T i T i 1 P i q(q q 1 )P i Ti 2 P i 2(q q 1 )P i T i P i +(q q 1 ) 2 P i ) (by (A2)) = q 2 (qp i T i 1 T i T i 1 P i q(q q 1 ) 2 P i T i P i q(q q 1 )P i 2(q q 1 )P i T i P i +(q q 1 ) 2 P i (by (A1)) = q 2 (q 3 P i T i P i q(q q 1 ) 2 P i T i P i q(q q 1 )P i 2(q q 1 )P i T i P i +(q q 1 ) 2 P i ) (by (E4 )) = q 2 ((q 3 q(q q 1 ) 2 2(q q 1 ))P i T i P i +(q q 1 )( q + q q 1 )P i ) = q(p i T i P i (q q 1 )P i ) = P i+1 Finally, we prove (E4) First let j<i Then, by (E4 ), P i+1 T j = q(p i T i P i T j (q q 1 )P i T j )=q(qp i T i P i (q q 1 )qp i )=qp i+1,

8 8 tom halverson and arun ram and T j P i+1 = qp i+1 is similar Now we consider the case where j = i, P i+1 T i = q(p i T i P i T i (q q 1 )P i T i ) (by (E5)) = q(p i T i qp i 1 T i 1 P i 1 T i q(q q 1 )P i T i P i 1 T i (q q 1 )P i T i ) (by (E5)) = q(qp i P i 1 T i T i 1 T i P i 1 q(q q 1 )P i P i 1 Ti 2 (q q 1 )P i T i ) (by (E1)) = q(qp i T i T i 1 T i P i 1 q(q q 1 )P i Ti 2 (q q 1 )P i T i ) (by (E2)) = q(qp i T i 1 T i T i 1 P i 1 q(q q 1 )P i Ti 2 (q q 1 )P i T i ) (by (A2)) = q(q 2 P i T i T i 1 P i 1 q(q q 1 )P i Ti 2 (q q 1 )P i T i ) (by (E4 )) = q(q 2 P i T i T i 1 P i 1 q(q q 1 ) 2 P i T i q(q q 1 )P i (q q 1 )P i T i ) (by (A1)) = q(q 2 P i T i T i 1 P i 1 q 2 (q q 1 )P i T i q(q q 1 )P i ) = q(q 2 P i P i 1 T i T i 1 P i 1 q 2 (q q 1 )P i T i q(q q 1 )P i ) (by (E2)) = q(q 2 P i T i P i 1 T i 1 P i 1 q 2 (q q 1 )P i T i q(q q 1 )P i ) (by (E1)) = q(qp i T i P i + q 2 (q q 1 )P i T i P i 1 q 2 (q q 1 )P i T i q(q q 1 )P i ) (by (E5)) = q(p i+1 + q 2 (q q 1 )P i T i P i 1 q 2 (q q 1 )P i T i ) (by (E5)) = q(p i+1 + q 2 (q q 1 )P i P i 1 T i q 2 (q q 1 )P i T i ) (by (E1)) = q(p i+1 + q 2 (q q 1 )P i T i q 2 (q q 1 )P i T i ) (by (E2)) = qp i+1 The case T i P i+1 = qp i+1 is similar Propositions 15 and 19 give the following theorem Theorem 110 R k (q) = A k (q) and thus (16) is a new presentation of R k (q) 2 Hecke algebras The affine Hecke algebra H k Fix q C The affine Hecke algebra H k is the algebra given by generators with relations X 1,,X k and T 1,,T k 1 (1) T i T j = T j T i, i j > 1, (2) T i T i+1 T i = T i+1 T i T i+1, 1 i k 2, (3) Ti 2 =(q q 1 )T i +1, 1 i k, (4) X i X j = X j X i, 1 i, j k, (5) X i T i = T i X i+1 +(q q 1 )X i, 1 i k 1 It follows from relations (3) and (5) that X i = T i 1 T 2 T 1 X 1 T 1 T 2 T i 1, for 1 i k,

9 and from (4) that (6) X 1 T 1 X 1 T 1 = T 1 X 1 T 1 X 1 rook monoid algebras and hecke algebras 9 In fact, Hk can be presented as the algebra generated by X 1 and T 1,,T k 1 with relations (1-3) and (6) Let [k]! = [1][2] [k], where [i] =1+q q 2(i 1) When [k]! 0 a large class of irreducible representations of the affine Hecke algebra H k (q), the integrally calibrated irreducible representations, have a simple combinatorial construction An H k - module M is integrally calibrated if M has a basis of simultaneous eigenvectors for X 1,X 2,,X k for which the eigenvalues are all of the form q j with j Z The construction of these H k -modules is originally due to Cherednik [Ch] (see [Ra] for greater detail) and is a generalization of the classical seminormal construction of the irreducible representations of the symmetric group by A Young Young s construction had been generalized to Iwahori-Hecke algebras of classical type (see Theorem 28 below) by Hoefsmit [Ho] in 1974 To describe the construction we shall use the notations of [Mac] for partitions so that a partition is identified with a collection of boxes in a corner, l(λ) is the number of rows of λ, and λ is the number of boxes in λ For example, the partition λ =(5, 5, 3, 1, 1) = has l(λ) =5and λ = 15 If λ is a partition that is obtained from µ by adding k boxes let λ/µ be the skew shape consisting of those boxes of λ that are not in µ Astandard tableau of shape λ/µ is a filling of the boxes of λ/µ with 1, 2,,k such that (a) the entries in the rows increase left to right, and (b) the entries in the columns increase top to bottom If b isaboxinλ/µ define CT(b) =q 2(c r), if b is in position (r, c) ofλ (21) Theorem 22 ([Ch], see also [Ra, Theorem 41], and [OR, Theorem 620a]) Assume that [k]! 0 Then the calibrated irreducible representations H λ/µ of the affine Hecke algebra H k are indexed by skew shapes and can be given explicitly as the vector space H λ/µ = C-span{v L L is a standard tableau of shape λ/µ} (so that the symbols v L form a basis of H λ/µ ) with H k -action given by X i v L = CT(L(i))v L, ( CT(L(i + 1))(q q 1 ) ( ) T i v L = v L + q 1 + CT(L(i + 1))(q ) q 1 ) v si L, CT(L(i + 1)) CT(L(i)) CT(L(i + 1)) CT(L(i)) where s i L is the same as L except i and i +1are switched, and

10 10 tom halverson and arun ram v si L =0,ifs i L is not a standard tableau Remark 23 In [OR] it is explained how the basis v L of H λ/µ and the action of H k in Theorem 22 can be derived in a natural way from the general mechanism of quantum groups (R-matrices, quantum Casimirs, the tensor product rule in (31)) and a Schur-Weyl duality theorem (Theorem 33 below) for the affine Hecke algebra The cyclotomic Hecke algebra H k (u 1,,u r ; q) Let u 1,,u r C and q C The cyclotomic Hecke algebra H k (u 1,,u r ; q) is the quotient of the affine Hecke algebra H k by the ideal generated by the relation (X 1 u 1 )(X 1 u 2 ) (X 1 u r )=0 (24) The algebra H k (u 1,,u r ; q) is a deformation of the group algebra of the complex reflection group G(r, 1,k)=(Z/rZ) S k and is of dimension dim(h k (u 1,,u r ; q)) = r k k! These algebras were introduced by Ariki and Koike [AK] Ariki has generalized the classical result of Gyoja and Uno [GU] and given precise conditions for the semisimplicity of the cyclotomic Hecke algebras Theorem 25 ([Ar]) The algebra H k (u 1,,,u r ; q) is semisimple if and only if q 2d u i u j for all k <d<k, 1 i<j r, and [k]! 0, where [k]! = [1][2] [k] and [i] =1+q q 2(i 1) Proof Let us only explain the conversion between the statement in [Ar] and the statement here This conversion is the same as in Remark 12 If Ti = qt i then T 2 i = q 2 ((q q 1 )T i +1) = (q 2 1)qT i + q 2 =(q 2 1) T i + q 2, which shows that our algebra is the same as Ariki s except with parameter q 2 Ariki and Koike give a combinatorial construction of the irreducible representations of the cyclotomic Hecke algebra H k (u 1,,u r ; q) when it is semisimple Define Ĥ (r) k = {r-tuples λ =(λ (1),,λ (r) ) of partitions with k boxes total} (26) Let λ Ĥ(r) k Astandard tableau of shape λ is a filling of the boxes of λ with 1, 2,,k such that for each λ (i),1 i r, (a) the entries in the rows are increasing left to right, and (b) the entries in the columns are increasing top to bottom Let L(i) denote the the box of L containing i, and define CT(b) =u i q 2(c r), if the box b is in position (r, c) ofλ (i) (27) Theorem 28 [AK, Theorem 37] If H k (u 1,u 2,,u r ; q) is semisimple its irreducible representations H λ, λ Ĥ(r) k, are given by H λ = H (λ(1),,λ (r)) = C-span{v L L is a standard tableau of shape λ}

11 rook monoid algebras and hecke algebras 11 (so that the symbols v L form a basis of the vector space H λ ) with H k (u 1,,u r ; q)-action given by X i v L = CT(L(i))v L, and T i v L =(T i ) LL v L +(q 1 +(T i ) LL )v si L, where q, if L(i) and L(i +1)are in the same row, of λ (j) for some fixed j, q (T i ) LL = 1, if L(i) and L(i +1)are in the same column, of λ (j) for some fixed j, CT(L(i + 1))(q q 1 ) CT(L(i + 1)) CT(L(i)), otherwise, s i L is the same as L except i and i +1are switched, and v si L =0,ifs i L is not a standard tableau It is interesting to note that Theorem 28 is almost an immediate consequence of Theorem 22 The Iwahori-Hecke algebra H k (u 1,u 2 ; q) of type B k The Iwahori-Hecke algebra of type B k is the cyclotomic Hecke algebra H k (u 1,u 2 ; q) Thus, for u 1,u 2 C and q C, H k (u 1,u 2 ; q) is the quotient of the affine Hecke algebra by the ideal generated by the relation (X 1 u 1 )(X 1 u 2 )=0 (29) The algebra H k (1, 1; q) is the group algebra of the Weyl group of type B k (the hyperoctahedral group of signed permutations) In the case of the Iwahori-Hecke algebra H k (u 1,u 2 ; q) oftypeb k Theorem 28 is due to Hoefsmit [Ho] When H k (u 1,u 2 ; q) is semisimple Hoefsmit s construction of the irreducible representations of H k (u 1,u 2 ; q) implies that, as H k 1 (u 1,u 2 ; q)-modules Res H k H k 1 H λ = λ H λ, (210) where the sum runs over all pairs of partitions λ which are obtained from λ by removing a single box, and { } L is a standard tableau of shape λ H λ = C-span v L and L has shape λ, (211) where L is the standard tableau with k 1 boxes which is obtained by removing the entry k from L The restriction rules (210) can be encoded in the Bratteli diagram for the sequence of algebras ie, the graph which has H 1 (u 1,u 2 ; q) H 2 (u 1,u 2 ; q) H 3 (u 1,u 2 ; q) (212) vertices on level k indexed by λ Ĥ(2) k and edges λ λ if λ is obtained from λ by removing a single box The first few rows of the Bratteli diagram for H k (u 1,u 2 ; q) are displayed in Figure 1

12 12 tom halverson and arun ram (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Figure 1 Bratteli Diagram for H k (u 1,u 2 ; q) If H k (u 1,u 2 ; q) is semisimple then H k (u 1,u 2 ; q) = M dλ (C), (213) λ ˆB k where d λ is the number of standard tableaux L of shape λ, and M d (C) is the algebra of d d matrices with entries from C The minimal ideals I λ of H k (u 1,u 2 ; q) are in one-to-one correspondence with the summands in (213) Let m<k, let I µ be a fixed minimal ideal of H m (u 1,u 2 ; q) and define I µ k is the ideal of H k (u 1,u 2 ; q) generated by I µ (I µ H m (u 1,u 2 ; q) H k (u 1,u 2 ; q)) The restriction rules (210) imply that I µ k = λ µ I λ, (214) where the sum is over all pairs of partitions λ =(λ (1),λ (2) ) H (2) k µ =(µ (1),µ (2) ) Ĥ(2) m by adding (k m) boxes which are obtained from The ideal I ((12 ), ) Lemma 215 Assume that H 2 (u 1,u 2 ; q) is semisimple The minimal ideal I ((12), ) of H 2 (u 1,u 2 ; q) is generated by the element { (X1 u p = 2 )(X 2 u 2 )(X 2 q 2 u 1 ), if u 1 0, where X (X 1 u 2 )(T 1 q)(x 1 u 2 )(X 2 u 2 ), if u 1 =0, 2 = T 1 X 1 T 1 Proof Using the construction of the simple H 2 (u 1,u 2 ; q)-modules in Theorem 28 it is not tedious to check that, when u 1 0, { (u1 u pv L = 2 )(q 2 u 1 u 2 )(q 2 u 1 q 2 u 1 )v L, if L has shape ((1 2 ), ), 0, otherwise, and, when u 1 =0, { (0 u2 )( q pv L = 1 q)(0 u 2 )(0 u 2 )v L, if L has shape ((1 2 ), ), 0, otherwise Thus p is an element of the ideal I ((12 ), ) Since I ((12 ), ) is a minimal ideal it is generated by any one of its (nonzero) elements

13 The algebra A k (u 1,u 2 ; q) rook monoid algebras and hecke algebras 13 Let u 1 C and u 2,q C Let A k (u 1,u 2 ; q) be the algebra given by generators and relations X 1 and T 1,T 2,,T k 1 (1) T i T j = T j T i, i j > 1, (2) T i T i+1 T i = T i+1 T i T i+1, 1 i n 2, (3) Ti 2 =(q q 1 )T i + q, 1 i k 1, (4) X 1 T 1 X 1 T 1 = T 1 X 1 T 1 X 1, (5) (X 1 u 1 )(X 1 u 2 )=0, (6) (X 1 u 2 )(X 2 u 2 )(X 2 q 2 u 1 )=0, if u 1 0, (X 1 u 2 )(T 1 q)(x 1 u 2 )(X 2 u 2 ), if u 1 = 0, where X 2 = T 1 X 1 T 1 Let  k = {(λ (1),λ (2) ) Ĥ(2) k λ (1) has at most one row} (216) Theorem 217 Assume H k (u 1,u 2 ; q) is semisimple (a) A k (u 1,u 2 ; q) is semisimple (b) As in (214) let I ((12), ) k be the ideal of H k (u 1,u 2 ; q) generated by the minimal ideal I ((12 ), ) of H 2 (u 1,u 2 ; q) Then A k (u 1,u 2 ; q) = H k(u 1,u 2 ; q) I ((12 ), ) k (c) As in (213) let d λ denote the number of standard tableaux of shape λ =(λ (1),λ (2) ) and M d (C) the algebra of d d matrices with entries from C Then A k (u 1,u 2 ; q) = λ Âk M dλ (C) (d) The irreducible A k (u 1,u 2 ; q)-modules H λ, λ Âk, are given by Hoefsmit s construction (Theorem 28) (e) The Bratelli diagram for the sequence of algebras A 1 (u 1,u 2 ; q) A 2 (u 1,u 2 ; q) A k (u 1,u 2 ; q) has vertices on level m indexed by λ Âm and edges λ λ if λ is obtained from λ by removing a box See Figure 2 Proof (a) follows from the fact that A k (u 1,u 2 ; q) is a quotient of H k (u 1,u 2 ; q) (b) By Lemma 215, the element p generates the ideal I ((12), ) in H k (u 1,u 2 ; q) and so this is a consequence of the definition of A k (u 1,u 2 ; q) (c) By (214) I ((12), ) k = I λ, (218) λ ((1 2 ), ) and so, by (b) and (213), the simple components of A k (u 1,u 2 ; q) are indexed by those elements of λ Ĥ(2) k which do not contain ((1 2 ), ) These are exactly the elements of Âk

14 14 tom halverson and arun ram (d) and (e) are conquences of (b), (c) and (218) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Figure 2 Bratteli Diagram of A k (u 1,u 2 ; q) and R k (q) By construction it is clear that the Bratteli diagram for A k (u 1,u 2 ; q) is a subgraph of the Bratteli diagram for H k (u 1,u 2 ; q) It is the subgraph which is obtained by removing all pairs of partitions λ =(λ (1),λ (2) ) which appear in (218) (for all k) Thus, it is the subgraph which is obtained by removing the vertex ((1 2 ), ) and all its descendants, ie, all pairs of partitions λ =(λ (1),λ (2) ) which are obtained by adding boxes to ((1 2 ), ) Theorem 219 The algebra A k (u 1,u 2 ; q) is semisimple if and only if q 2d u 1 u 2 for all k <d<k, and [k]! 0, where [k]! = [1][2] [k] and [i] =1+q q 2(i 1) Proof Since A k (u 1,u 2 ; q) is a quotient of H k (u 1,u 2 ; q), we know that A k (u 1,u 2 ; q) is semisimple when H k (u 1,u 2 ; q) is Thus A k (u 1,u 2 ; q) is semisimple when (a) and (b) hold The Iwahori-Hecke algebra H k (q) oftypea k 1 is the cyclotomic Hecke algebra H k (1, 1; q) Thus, H k (q) is the quotient A k (u 1,u 2 ; q) by the relation X 1 = u 1 By Theorem 25 (in this case orginally due to Gyoja and Uno) H k (q) is semisimple if and only if [k]! 0 Since H k (q) isa quotient of A k (u 1,u 2 ; q), the algebra A k (u 1,u 2 ; q) is not semisimple when H k (q) is not semisimple Thus, A k (u 1,u 2 ; q) is not semisimple when [k]!=0 If u 1 = u 2 then the representation ρ: A k (u 1,u 2 ; q) M 2 (C) given by setting ( ) u1 1 ρ(x 1 )= 0 u 1 and ρ(t i )= ( ) q 0 0 q is an indecomposable representation which is not irreducible Thus A k (u 1,u 1 ; q) is not semisimple The Tits deformation theorem (see [CR, (6817)]) says that the algebra A k (u 1,u 2 ; q) has the same structure for any choice of the parameters u 1,u 2 for which it is semisimple Assume that [k]! 0 and u 2 = q 2d u 1, u 1 0 Let λ/µ be the skew shape given by λ =(k 1,d) and µ =(d 1) and define CT(b) =u 1 q 2(c r)+2, if b is a box in position (r, c) ofλ With these definitions the formulas in Theorem 22 define an H k (u 1,q 2d u 1 ; q)-module H λ/µ A check that (X 1 q 2d u 1 )(X 2 q 2d u 1 )(X 2 q 2 u 1 )v L =0,

15 rook monoid algebras and hecke algebras 15 for all standard tableaux L of shape λ/µ shows that that H λ/µ is an A k (u 1,q 2d u 1 ; q)-module The standard proof (see [Ra, Theorem 41]) of Theorem 25 applies in this case to show that H λ/µ is an irreducible A k (u 1,q 2d u 1 ; q)-module It has dimension dim(h λ/µ ) = (the number of standard tableaux of shape λ/µ) = ( ) k 1 d When we restrict this A k (u 1,q 2d u 1 ; q)-module to H k (q), it is a direct sum of irreducible H k (q)- modules indexed by partitions ν k with multiplicity given by the classical Littlewood-Richardson coefficient c λ µν (see [Ra, Theorem 61]) Since λ =(k 1,d) and µ =(d 1) we have c λ µν 0 only if ν has 2rowsandc λ µν = 0 when ν =(k) So H λ/µ is an irreducible A k (u 1,q 2 du 1 ; q)-module such that upon restriction to H k (q) is a direct sum of irreducible representations indexed by partitions with length 2, and which does not contain the trivial representation of H k (q) When A k (u 1,u 2 ; q) is semisimple its irreducible representations H λ are indexed by pairs of parititions λ =(λ (1),λ (2) ) such that λ (1) has at most one row If λ (1) has length r, then, on restriction to H k (q), H λ is a direct sum of irreducibles indexed by partitions ν k with multiplicites c ν By the Pieri rule [Mac, (516)] the resulting ν are those obtained by adding a horizontal λ (1) λ (2) strip of length r to λ (2) Only when λ (2) has a single row will all the ν have 2 rows and, in this case, c ν = c ν λ (1) λ (2) (r),(k r) = 1 for ν =(k) Thus, when A k(u 1,u 2 ; q) is semisimple every irreducible representation which, on restriction to H k (q), decomposes as a direct sum of components indexed by partitions with 2 rows does contain the trivial representation of H k (q) Thus, the Tits deformation theorem implies that A k (u 1,q 2d u 1 ; q) is not semisimple Remark 220 It is interesting to note that the blob algebras (see [MW]) are also quotients of A k (u 1,u 2 ; q) The q-rook monoid algebras R k (q) The new presentation of the q-rook monoid given in Section 1 shows that R k (q) =A k (0, 1; q), and thus R k (q) is a quotient of Iwahori-Hecke algebra H n (0, 1; q) oftypeb k Corollary 221 (a) The q-rook monoid algebra R k (q) is semisimple if and only if [k] q! 0 (b) If R k (q) is semisimple then the irreducible representations of R k (q) are indexed by λ Âk (see (216)) and are given explicitly by the construction in Theorem 28 Part (a) of Corollary 221 is Theorem 219 applied to R k (q) and (b) is Theorem 217(e) for R k (q) Part (a) was proved in a different way by Solomon [So4] and part (b) is the result of Halverson [Ha, Theorem 32] which was the catalyst for the results of this paper As in (210) it follows that, as R k 1 (q)-modules Res R k R k 1 R λ = λ R λ,

16 16 tom halverson and arun ram where the sum runs over all pairs of partitions λ which are obtained from λ by removing a single box, and R λ = C-span{v L L has shape λ }, where L is the standard tableau with k 1 boxes which is obtained by removing the k from L The first few rows of the Bratteli diagram for the sequence of algebras are as displayed in Figure 2 3 Schur-Weyl dualities R 1 (q) R 2 (q) R 3 (q) Let U q gl(n) be the quantum group corresponding to GL n (C) This is the algebra given by generators e i, f i, (1 i<n), and q ±ε i, (1 i n), with relations q ε i q ε j = q ε j q ε i, q ε i q ε i = q ε i q ε i =1, q 1 e j, if j = i 1, qf j, if j = i 1, q ε i e j q ε i = qe j, if j = i, q ε i f j q ε i = q 1 f j, if j = i, e j, otherwise, f j, otherwise, e i f j f j e i = δ ij q ε i ε i+1 q (ε i ε i+1 ) q q 1, e i±1 e 2 i (q + q 1 )e i e i±1 e i + e 2 i e i±1 =0, f i±1 f 2 i (q + q 1 )f i f i±1 f i + f 2 i f i±1 =0, e i e j = e j e i, f i f j = f j f i, if i j > 1 Part of the data of a quantum group is an R-matrix, which provides a canonical U q gl(n)-module isomorphism Ř MN : M N N M for any two U q gl(n)-modules M and N The irreducible polynomial representations L(λ) ofu q gl(n) are indexed by dominant integral weights λ L + where L = n Zε i = {λ = λ 1 ε λ n ε n λ i Z}, i=1 and L + = {λ L λ 1 λ 2 λ n } The elements of L + can be identified with partitions λ with n rows The irreducible representation V = L(ε 1 ) has dim(v )=n, and L(µ) V = L(µ + ) (31) µ +

17 rook monoid algebras and hecke algebras 17 as U q gl(n)-modules, where the direct sum is over all partitions µ + which are obtained from µ by adding a box The U q gl(n)-module V can be given explicitly as the vector space V = C-span{v 1,,v n } (so that the symbols v i form a basis of V ) with U q gl(n)-action given by e i v j = { vj 1, if j = i +1, 0, if j i +1, q ±ε i v j = { vj+1, if j = i, f i v j = 0, if j i, { q ±1 v j, if j = i, v j, if j i and With this notation the R-matrix for V V is given explicitly by Ř VV : V V V V, where qv j v i, if i = j, Ř VV (v i v j )= v j v i, if i>j, v j v i +(q q 1 )(v i v j ), if i<j (32) A Schur-Weyl duality for affine and cyclotomic Hecke algebras Theorem 33 (see [OR, Theorem 617ab and Theorem 618]) (a) For any µ L + there is an action of the affine Hecke algebra H k on L(µ) V k given by Φ: H k End(L(µ) V k ) where Φ(X 1 )=ŘV,L(µ)ŘL(µ),V id (k 1) V and Φ(T i )=id L(µ) id (i 1) V ŘVV id (k i 1) V (b) The H k action on L(µ) V k commutes with the U q gl(n)-action and the map (c) As a (U q gl(n), H k ) bimodule Φ: H k End Uq gl(n)(l(µ) V k ) is surjective L(µ) V k = L(λ) H λ/µ, where the sum is over all partitions λ which are obtained from µ by adding k boxes and H λ/µ is a simple H k -module (d) The representation Φ given in part (a) is a representation of the cyclotomic Hecke algebra H k (u 1,,u r ; q), ie Φ: H k (u 1,,u r ; q) End Uq gl(n)(l(µ) V k ), for any (multi)set of parameters u 1,,u r containing the (multi)set of values CT(b) (defined in (21)) as b runs over the boxes which can be added to µ (to get a partition) λ Remark 34 The affine Hecke algebra module H λ/µ which appears in Theorem 33(c) is the same as the module H λ/µ constructed in Theorem 22

18 18 tom halverson and arun ram A Schur-Weyl duality for Iwahori-Hecke algebras of type B Suppose that µ is a partition with two addable boxes, ie { µ = d = l d, for some 0 <d n, l Z 0 } {{ } l Proposition 35 Let µ = l d, 0 <d n, l Z 0 Then Theorem 33(d) provides a Schur-Weyl duality for H k (u 1,u 2 ; q) with u 1 = q 2l and u 2 = q 2d Proof One only needs to note that if b 1,b 2 are the addable boxes of µ and CT(b) is as defined in (21) then u 1 = CT(b 1 )=q 2l, and u 2 = CT(b 2 )=q 2d A Schur-Weyl duality for A n (u 1,u 2 ; q) Keeping the notation of Proposition 35, consider the special case l = n 1 and d k, so that { µ = n 1 = l (n 1), for some l Z 0 } {{ } l Then, as a (U q gl(n),h k (u 1,u 2 ; q) bimodule L(µ) V k = L(λ) H λ, where λ = λ µ λ (1) λ (2) (36) and H λ is a simple H k (u 1,u 2 ; q) module indexed by a pair of partitions λ =(λ (1),λ (2) ) with k boxes total and such that λ (1) has at most one row The following result shows that, in this case, the Schur-Weyl duality in Theorem 33 becomes a Schur-Weyl duality for the algebra A k (u 1,u 2 ; q), where u 1 = q 2(n 1) and u 2 = q 2l Proposition 37 Let µ =(n 1) d Then the H k action on L(µ) V k which is given by Theorem 33 factors through the algebra A k (u 1,u 2 ; q), where u 1 = q 2(n 1) and u 2 = q 2l Proof By Theorem 33(d), the H k action on L(µ) V k factors through the algebra H k (u 1,u 2 ; q) where u 1 = q 2(n 1) and u 2 = q 2l It remains to check that (X 1 u 2 )(X 2 u 2 )X 2 q 2 u 1 )=0as operators on L(µ) V k To do this it is sufficient to show that (X 1 u 2 )(X 2 u 2 )X 2 q 2 u 1 )=0as operators on H λ for each H λ which appears in the decomposition (36) The H k (u 1,u 2 ; q)-module H λ has basis indexed by the standard tableaux L of shape λ and (X 1 u 2 )(X 2 u 2 )(X 2 q 2 u 1 )v L =(CT(L(1)) u 2 )(CT(L(2)) u 2 )(CT(L(2)) q 2 u 1 )v L For each of the possible positions of the first two boxes of L at least one of the factors in the last product is 0 Thus (X 1 u 2 )(X 2 u 2 )X 2 q 2 u 1 ) = 0 as operators on H λ

19 rook monoid algebras and hecke algebras 19 Another Schur-Weyl duality for cyclotomic Hecke algebras Let m 1,,m r be positive integers such that m 1 + m r = n Then g P = gl(m 1 ) gl(m r ) is a Lie subalgebra of gl(n), and correspondingly U P = U q gl(m 1 ) U q gl(m r ) is a subalgebra of U q gl(n) There is a corresponding decomposition of the fundamental representation V of U q gl(n) asau P - module: V = V 1 V r, where dim(v j )=m j, and V j is the fundamental representation for U q gl(m j ) and say that v is homogeneous of degree j Let u 1,,u r C and define If v V j, we write deg(v) =j d: V V by d(v) =u j v, if deg(v) =j (38) Recall the action of ŘVV: V V V V as given in (32), define { ŘVV (v w), if deg(v) = deg(w), Š VV : V V V V by Š VV (v w) = w v, if deg(v) deg(w), (for homogeneous v, w V ), and define d i,r i,s i End(V k )by d i =id (i 1) V Ř i =id (i 1) V Š i =id (i 1) V d id (k i) V, 1 i k, ŘVV id (k i 1) V, 1 i k 1, ŠVV id (k i 1) V, 1 i k 1 Theorem 310 (Sakamoto-Shoji [SS]) (a) There is an action of H k (u 1,u 2,,u r ; q) on V k given by Φ P : H k (u 1,,u r ; q) End(V k ) where Φ P (T i )=Ři and Φ P (X 1 )=Ř 1 1 Ř 1 k Šk Š1d 1 (b) The action of H k (u 1,,u r ; q) commutes with the action of U P on V k, ie, Φ P : H k (u 1,,u k ; q) End UP (V k ) (c) As a (U P,H k (u 1,,u r ; q))-bimodule V k = L P (λ) H λ, λ=(λ (1),,λ (r) ) where the sum is over all r-tuples λ =(λ (1),,λ (r) ) of partitions such that l(λ (j) ) m j, L P (λ) is the simple U P -module given by L P (λ) =L (1) (λ (1) ) L (r) (λ (r) ), where L (j) (λ (j) ) is the simple U q (gl(m j ))-module correspnding to the partition λ (j), and H λ is a (not necessarily simple) H k (u 1,,u r ; q)-module (39) Remark 311 The H k (u 1,,u k ; q)-module H λ appearing in Theorem 310(c) is simple whenever H k (u 1,,u k ; q) is semisimple In that case H λ coincides with the H k (u 1,,u r ; q)-module constructed in Theorem 28

20 20 tom halverson and arun ram A Schur-Weyl duality for R k (q) Consider the case of Theorem 310 when r =2, m 1 =1, m 2 = n, u 1 =0, u 2 =1 Then V = V 1 V 2 where V 1 = Cv 0, and V 2 = C-span{v 1,,v n } Let us analyze the action of H k (0, 1; q) onv k as given by Sakamoto and Shoji Since dv 0 = u 1 v 0 =0, Φ P (X 1 )(v 0 v i2 v ik )=0, and, for l>0, Φ P (X 1 )(v l v i2 v )=Ř 1 ik 1 Ř 1 k 1Šk 1 Š1d 1 (v l v i2 v ik ) = Ř 1 1 Ř 1 k 1Šk 1 Š1(v l v i2 v ik ) = Ř 1 1 Ř 1 k 1Řk 1 Ř1(v l v i2 v ik ) = v l v i2 v ik, and thus Φ P (X 1 ) = d 1 This calculation shows that the Sakamoto-Shoji action of H k (0, 1; q) coincides exactly with action for the Schur-Weyl duality for the q-rook monoid algebra R k (q) in the form given by Halverson [Ha, Corollary 63] 4 References [AK] S Ariki and K Koike, A Hecke algebra of (Z/rZ) S n and construction of its irreducible representations, Adv Math 106 (1994), [Ar] S Ariki, On the semisimplicity of the Hecke algbera of (Z/rZ) S n, J Algebra 169 (1994), [Ch] I Cherednik, A new interpretation of Gelfand-Tzetlin bases, Duke Math J 54 (1987), [CR] C Curtis and I Reiner, Methods of Representation Theory: With Applications to Finite Groups and Orders, Volume II, Wiley, New York, 1987 [GU] A Gyoja and K Uno, On the semisimplicity of Hecke algebras, J Math Soc Japan 41 (1989), [Ha] T Halverson, Representations of the q-rook monoid, preprint (2001) [Ho] P N Hoefsmit, Representations of Hecke algebras of finite groups with BN-pairs of classical type, Thesis, Univ of British Columbia, 1974 [Mac] I G Macdonald, Symmetric Functions and Hall polynomials, Second edition, Oxford University Press, New York, 1995 [MW] P P Martin and D Woodcock, On the structure of the blob algebra, J Algebra 225 (2000), [Mu1] W D Munn, Matrix representations of semigroups, Proc Camb Phil Soc, 53 (1957), 5-12

21 rook monoid algebras and hecke algebras 21 [Mu2] W D Munn, The characters of the symmetric inverse semigroup, Proc Camb Phil Soc, 53 (1957), [OR] R Orellana and A Ram, Affine braids, Markov traces and the category O, preprint (2001) [Ra] A Ram, Skew shape representations are irreducible, preprint (1998) [So1] L Solomon, The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field, Geom Dedicata 36 (1990), [So2] L Solomon, Representations of the rook monoid, J Algebra, to appear [So3] L Solomon, Abstract No , Abstracts Presented to the American Math Soc, Vol 16, No 2, Spring 1995 [So4] L Solomon, The Iwahori algebra of M n (F q ), a presentation and a representation on tensor space, preprint (2001) [SS] S Sakamoto and T Shoji, Schur-Weyl reciprocity for Ariki-Koike algebras, J Algebra 221 (1999), [Yg] A Young, On quantitative substitutional analysis VI, Proc London Math Soc 31 (1931),