Table Of Contentq-rook monoid algebras, Hecke algebras, and Schur-Weyl duality
4 Tom Halverson∗
0 Department of Mathematics and Computer Science
0
Macalester College
2
St. Paul, MN 55105
n [email protected]
a
J
and
3
2 Arun Ram∗∗
Department of Mathematics
]
T University of Wisconsin, Madison
R
Madison, WI 53706
. [email protected]
h
t
a
m
[ In memory of Sergei Kerov 1946-2000
1
When we were at the beginnings of our careers Sergei’s support helped us to
v
0 believe in our work. He generously encouraged us to publish our results on
3
Brauer and Birman-Murakami-Wenzl algebras, results which had in part, or
3
possibly in total, been obtained earlier by Sergei himself. He remains a great
1
0 inspiration for us, both mathematically and in our memory of his kindness,
4 modesty, generosity, and encouragement to the younger generation.
0
/
h
t
a
0. Introduction
m
: The rook monoid R is the monoid of k ×k matrices with entries from {0,1} and at most
v k
one nonzero entry in each row and column. Recently, the representation theory of its “Iwahori-
i
X
Hecke” algebra R (q), called the q-rook monoid algebra, has been analyzed. In particular, a
k
r Schur-Weyltypedualityontensorspacewasfoundfortheq-rookmonoidalgebraanditsirreducible
a
representationsweregivenexplicitcombinatorialconstructions. Inthispaperweshowthat,infact,
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 (q) by
k
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 con-
struction of irreducible representations of cyclotomic Hecke algebras by Ariki and Koike
∗
ResearchsupportedinpartbyNationalScience FoundationgrantDMS-0100975.
∗∗
Research supported in part by National Science Foundation grant DMS-9971099,the National Security Agency
andEPSRCgrantGRK99015.
2 tom halverson and arun ram
[AK], and the constructionof the irreducible representationsof Iwahori-Hecke algebrasof
type B by Hoefsmit [Ho].
(b) The Schur-Weyl duality for theq-rook monoid algebradiscovered 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)forthefinitealgebraicmonoidM (F ) ofn×nmatricesover afinitefield with q elements,
n q
with respect to the “Borel subgroup” B of invertible upper triangular matrices. Later Solomon
[So2] found a Schur-Weyl duality for R (1) in which R (1) acts as the centralizer algebra for the
k k
action of the general linear group GL (C) on V⊗k where V = L(ε )⊕L(0) is the direct sum of
n 1
the “fundamental” n-dimensional representation and the trivial module L(0) for GL (C). Then
n
Solomon [So3,4] gave a presentation of R (q) by generators and relations and defined an action of
k
R (q) on tensor space.
k
Halverson [Ha] found a new presentation of R (q) and used it to show that Solomon’s action
k
of R (q) on tensor space extends the Schur-Weyl duality so that R (q) is the centralizer of the
k k
quantumgenerallinear groupU gl(n) onV⊗k wherenow V = L(ε )⊕L(0)is thedirectsumof the
q 1
“fundamental”andthetrivialmoduleforU gl(n). Halversonalsoexploitedhisnewpresentationto
q
construct, combinatorially, all the irreducible representations of R (q) when R (q) is semisimple.
k k
The main results of this paper are the following:
(a) We find yet another presentation (1.6) of R (q) by generators and relations.
k
(b) Our new presentation shows that
R (q)= H (0,1;q)/I,
k k
where H (0,1;q) is the Iwahori-Hecke algebra of type B with parameters specialized to
k k
0 and 1, and I is the ideal generated by the minimal ideal of H (0,1;q) corresponding to
2
the pair of partitions λ =((12),∅).
(c) We show that the irreducible representations of R (q) found in [Ha] come from the con-
k
structions of irreducible representations of H (0,1;q).
k
(d) We usethe factthatR (q)is a quotient ofH (0,1;q) and thefactthatthe Iwahori-Hecke
k k
algebra H (q) of type A is a quotient of R (q) to easily determine, in Corollary 2.21,
k k−1 k
the values of q for which R (q) is semisimple. These values were first found in [So4] using
k
other methods.
(e) We show that the Schur-Weyl duality between R (q) and U gl(n) comes from the Schur-
k q
Weyl duality of Sakamoto and Shoji [SS] for the cyclotomic Hecke algebras (Theorem
3.5).
(f) We give a different Schur-Weyl dualityfor algebrasA (u ,u ;q) = H (u ,u ;q)/I, where
k 1 2 k 1 2
u ,u 6= 0, H (u ,u ;q) is the Iwahori-Hecke algebra of type B and I is the ideal
1 2 k 1 2 k
generated by the minimal ideal of H (u ,u ;q) corresponding to the pair of partitions
2 1 2
λ = ((12),∅). This Schur-Weyl duality comes from the Schur-Weyl duality of Orellana
and Ram for the affine Hecke algebra (Theorem 3.3).
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 (q) given by generators
k
P ,P ,...,P and T ,T ,...,T
1 2 k 1 2 k−1
with relations
(A1) T2 = (q−q−1)T +1, 1 ≤i ≤ k−1,
i i
(A2) T T T =T T T , 1 ≤i ≤ k−2,
i i+1 i i+1 i i+1
(A3) T T =T T , |i−j| >1,
i j j i
(R1) P2 = P , 1 ≤i ≤ k,
i i (1.1)
(R2) P P = P P , 1 ≤i,j ≤k,
i j j i
(R3) P T =T P , 1 ≤i < j ≤k,
i j j i
(R4) P T =T P =qP , 1 ≤j <i ≤k,
i j j i i
(R5) P =qP T−1P =q(P T P −(q−q−1)P ), 1 ≤i ≤ k−1.
i+1 i i i i i i i
The algebra R (q) was introduced by Solomon [So1] as an analogue of the Iwahori-Hecke
k
algebra for the finite algebraic monoid M (F ) of k×k matrices over a finite field with q elements
k q
with respect to its “Borel subgroup” of invertible upper triangular matrices. The presentation of
R (q) given above is due to Halverson [Ha].
k
When q = 1, R (q) specializes to the algebra of the rook monoid R that consists of k ×k
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 becomesthe matrix obtained by switching rows i and i+1 in the identity matrix
i
I and, for 1 ≤ i ≤k−1, P becomes the matrix E +E +···+E , where E is the
i i+1,i+1 i+1,i+2 k,k i,j
matrix with a 1 in position (i,j) and zeros elsewhere. The generatorP specializes to the 0 matrix
k
(which is not the 0 element in the monoid algebra).
Remark 1.2. ThedefinitionofR (q)in [So1,3,4]and[Ha] usesgeneratorsT˜ in placeofT . These
k i i
generators satisfy T˜2 =(q−1)T˜ +q in place of (A1). In our presentation, if we let T˜ =qT , then
i i i i
T˜2 = q2((q−q−1)T +1) =(q2−1)qT +q2 =(q2−1)T˜ +q2, which shows that our algebra is the
i i i i
same except with parameter q2 instead of q.
Define
X = T T ···T (1−P )T T ···T , 1 ≤i ≤k, (1.3)
i i−1 i−2 1 1 1 2 i−1
so that X =T X T .
i+1 i i i
Lemma 1.4. In R (q) we have the following relations
k
(a) P P = P P =P , for i ≤j.
i j j i j
(b) P X = P −P .
1 2 1 2
Proof. (a) If i =j this is (R1). If i< j then, by (R5) and induction,
P P =P (P T P −(q−q−1)P ) = P T P −(q−q−1)P =P .
i j i j−1 j−1 j−1 j−1 j−1 j−1 j−1 j−1 j
4 tom halverson and arun ram
(b) We use relations (A1), (R4), and (R5) to get
P X =P T (1−P )T =P T2−P T P T
1 2 1 1 1 1 1 1 1 1 1 1
=(q−q−1)P T +P −P T P T
1 1 1 1 1 1 1
=(q−q−1)P T +P −q−1P T −(q−q−1)P T
1 1 1 2 1 1 1
=P −q−1P T =P −P .
1 2 1 1 2
Proposition 1.5. Theq-rookmonoidalgebraR (q)isgeneratedbytheelementsX ,T ,...,T ,
k 1 1 k−1
and these elements satisfy the relations (A1), (A2), (A3), and
(B1) X T = T X , for 2 ≤j ≤ k,
1 j j 1
(B2) X2 = X ,
1 1
(B3) X T X T =T X T X ,
1 1 1 1 1 1 1 1
(B4) (1−X )(T −q)(1−X )(1−X ) =0, where X = T X T .
1 1 1 2 2 1 1 1
Proof. By (R5), R (q) is generated by X =1−P ,T ,...,T .
n 1 1 1 k−1
(B1) By (R3), X T =(1−P )T = T (1−P ) =T X .
1 j 1 j j 1 j 1
(B2) By (R1), X2 =(1−P )2 =1−2P +P2 =1−P =X .
1 1 1 1 1 1
(B3) Using (R4) and (R5),
q(T P T P −(q−q−1)T P ) =T P =P T =q(P T P T −(q−q−1)P T ),
1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1
and so
P T P T = T P T P +(q−q−1)(P T −T P ). (∗)
1 1 1 1 1 1 1 1 1 1 1 1
Now, using (∗) and (A1),
X T X T =(1−P )T (1−P )T
1 1 1 1 1 1 1 1
=T2 −P T2−T P T +P T P T
1 1 1 1 1 1 1 1 1 1
=T2 −(q−q−1)P T −P −T P T +T P T P +(q−q−1)(P T −T P )
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
=T2 −P −T P T +T P T P −(q−q−1)T P
1 1 1 1 1 1 1 1 1 1 1
=T2 −T2P −T P T +T P T P
1 1 1 1 1 1 1 1 1 1
=T (1−P )T (1−P )
1 1 1 1
=T X T X .
1 1 1 1
Finally, to show (B4), we use Lemma 1.4(b),
(1−X )(T −q)(1−X )(1−X ) =P (T −q)P (1−X )
1 1 1 2 1 1 1 2
=P T P −qP −P T P X +qP X
1 1 1 1 1 1 1 2 1 2
=P T P −qP −P T (P −P )+q(P −P )
1 1 1 1 1 1 1 2 1 2
=−qP +P T P
2 1 1 2
=−qP +qP P by (R4)
2 1 2
=−qP +qP by Lemma 1.4(a)
2 2
=0.
rook monoid algebras and hecke algebras 5
Define a new algebra A (q) by generators
k
X and T ,T ,...,T
1 1 2 k−1
and relations
(A1) T2 = (q−q−1)T +1, 1 ≤i ≤k−1,
i i
(A2) T T T =T T T , 1 ≤i ≤k−2,
i i+1 i i+1 i i+1
(A3) T T =T T , |i−j| >1,
i j j i
(B1) X T =T X , 2 ≤j ≤ k, (1.6)
1 j j 1
(B2) X2 =X ,
1 1
(B3) X T X T = T X T X ,
1 1 1 1 1 1 1 1
(B4) (1−X )(T −q)(1−X )(1−X ) =0, where X =T X T .
1 1 1 2 2 1 1 1
We will show that R (q) ∼=A (q). Define
k k
P =(1−X ) and P =q(P T P −(q−q−1)P ), 1 ≤ i ≤k−1. (1.7)
1 1 i+1 i i i i
Lemma 1.8. Relations (B1)-(B4) are equivalent, respectively, to
(B1′) P T = T P , for 2 ≤j ≤ n,
1 j j 1
(B2′) P2 =P ,
1 1
(B3′) P T =T P ,
2 1 1 2
(B4′) P2 =P .
2 2
Proof. Subtracting T from each side of
j
T −P T =(1−P )T =X T =T X =T (1−P ) =T −T P ,
j 1 j 1 j 1 j j 1 j 1 j j 1
shows that (B1) is equivalent to (B1′). Relations (B2) and (B2′) are equivalent since
1−P =X = X2 =(1−P )2 = 1−2P +P2.
1 1 1 1 1 1
Since
X T X T =(1−P )T (1−P )T =(1−P )T2−T P T +P T P T
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
=(1−P )((q−q−1)T +1)−T P T +(q−1P +(q−q−1)P )T (by (A1) and (1.7))
1 1 1 1 1 2 1 1
=(q−q−1)T +1−(q−q−1)P T −P −T P T +q−1P T +(q−q−1)P T
1 1 1 1 1 1 1 2 1 1 1
=(q−q−1)T +1−P −T P T +q−1P T
1 1 1 1 1 2 1
is equal to
T X T X =T (1−P )T (1−P ) =T2(1−P )−T P T +T P T P
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
=((q−q−1)T +1)(1−P )−T P T +T (q−1P +(q−q−1)P ) (by (A1) and (1.7))
1 1 1 1 1 1 2 1
=(q−q−1)T +1−(q−q−1)T P −P −T P T +q−1T P +(q−q−1)T P
1 1 1 1 1 1 1 1 2 1 1
=(q−q−1)T +1−P −T P T +q−1T P ,
1 1 1 1 1 1 2
(B3) is equivalent to (B3′).
6 tom halverson and arun ram
Expanding
(T −q)(1−X ) =(T −q)(1−T X T ) = (T −q)(1−T (1−P )T )
1 2 1 1 1 1 1 1 1 1
=(T −q)(1−(q−q−1)T −1+T P T ) (by (A1))
1 1 1 1 1
=(T −q)T (−(q−q−1)+P T )
1 1 1 1
=((q−q−1)T +1)−qT )(q−1−q+P T ) (by (A1))
1 1 1 1
=(1−q−1T )(q−1−q+P T )
1 1 1
=q−1 −q+P T −q−2T +T −q−1T P T
1 1 1 1 1 1 1
=−(q−q−1)+q−1(q−q−1)T +P T −q−1T P T ,
1 1 1 1 1 1
gives
(1−X )(T −q)(1−X )(1−X )= P (T −q)(1−X )P
1 1 2 1 1 1 2 1
= −(q−q−1)P +q−1(q−q−1)P T P +P T P −q−1P T P T P (by (B2′))
1 1 1 1 1 1 1 1 1 1 1 1
= −(q−q−1)P +(2−q−2)P T P −q−1(P T P )2 (by (B2′))
1 1 1 1 1 1 1
= −(q−q−1)P +(2−q−2)(q−1P +(q−q−1)P )−q−1(q−1P +(q−q−1)P )2 (by (1.7))
1 2 1 2 1
= (q−q−1)(−1+2−q−2)P +q−1(2−q−2)P
1 2
−q−1(q−2P2+2q−1(q−q−1)P +(q−q−1)2P )
2 2 1
= (q−q−1)2(q−1−q−1)P +q−1(q−2P −q−2P2)
1 2 2
= q−3(P −P2).
2 2
and so (1−X )(T −q)(1−X )(1−X ) = 0 if and only if P2 = P . Thus (B4) is equivalent to
1 1 2 1 2 2
(B4′).
Proposition 1.9. The algebra A (q) is generatedby T ,...,T ,P ,...,P , and these elements
k 1 k−1 1 k
satisfy the relations (A1), (A2), (A3), and
(E1) P T =T P , for 1 ≤ i <j ≤ k,
i j j i
(E2) P P =P P =P , for all 1 ≤j < i≤ k,
i j j i i
(E3) P2 =P , for 1 ≤i ≤ k,
i i
(E4) P T =T P = qP , for 1 ≤ j <i ≤ k,
i j j i i
(E5) P = qP T−1P = q(P T P −(q−q−1)P ), for 1 ≤i ≤k−1.
i+1 i i i i i i i
Proof. Since X = 1 − P , the elements T ,...,T ,P generate A (q). Relation (E5) is the
1 1 1 n−1 1 k
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 by induction and T commutes with T by (A3), so
i j i
T P =q−1T P T P −(q−q−1)T P =q−1P T P T −(q−q−1)P T =P T ,
j i+1 j i j i j i i j i j i j i+1 j
proving (E1).
We now prove (E2)-(E4) collectively by induction on i. The case i = 1 for (E2) follows from
(B2′) since
P P = q(P T P −(q−q−1)P )P = q(P T P −(q−q−1)P ) =P . (∗)
2 1 1 1 1 1 1 1 1 1 1 2
rook monoid algebras and hecke algebras 7
The relation P P =P is similar. The first two cases of (E3) are (B2′) and (B4′). The i= 1 case
1 2 2
of (E4) follows from (B4′) since
P2 =P q(P T P −(q−q−1)P )
2 2 1 1 1 1
=q(P T P −(q−q−1)P ) (by (∗))
2 1 1 2
=q(T P P −(q−q−1)P ) (by (B3′))
1 2 1 2
=q(T P −(q−q−1)P ) (by (∗)).
1 2 2
Since P2 = P , we have T P = qP . The case P T =qP is similar.
2 2 1 2 2 2 1 2
Now fix i >1 and assume the following relations,
(E2∗) P P =P P =P , for all 1 ≤j < i,
i j j i i
(E3∗) P2 =P ,
i i
(E4∗) P T =T P = qP , for 1 ≤ j <i,
i j j i i
We show each of these relations for i+1.
For (E2) we use (E3∗) to get
P P =q(P T P2−(q−q−1)P2) =q(P T P −(q−q−1)P ) =P ,
i+1 i i i i i i i i i i+1
and when j < i, we use (E2∗) to get
P P = q(P T P P −(q−q−1)P P )= q(P T P −(q−q−1)P ) =P .
i+1 j i i i j i j i i i i i+1
The relations P P =P and P P =P are similar, and so (E2) is established.
i i+1 i+1 j i+1 i+1
To establish (E3), let i >2 (note that we have established i= 1,2),
P2 =q2(P T P −(q−q−1)P )2
i+1 i i i i
= q2(P T P T P −2(q−q−1)P T P +(q−q−1)2P ) (by (E3∗))
i i i i i i i i i
= q2(P T qP T P T P −q(q−q−1)P T P T P
i i i−1 i−1 i−1 i i i i i−1 i i
−2(q−q−1)P T P +(q−q−1)2P ) (by (E5))
i i i i
= q2(qP P T T T P P −q(q−q−1)P P T2P
i i−1 i i−1 i i−1 i i i−1 i i
−2(q−q−1)P T P +(q−q−1)2P ) (by (E1))
i i i i
= q2(qP T T T P −q(q−q−1)P T2P −2(q−q−1)P T P +(q−q−1)2P ) (by (E2))
i i i−1 i i i i i i i i i
= q2(qP T T T P −q(q−q−1)P T2P −2(q−q−1)P T P +(q−q−1)2P ) (by (A2))
i i−1 i i−1 i i i i i i i i
= q2(qP T T T P −q(q−q−1)2P T P −q(q−q−1)P
i i−1 i i−1 i i i i i
−2(q−q−1)P T P +(q−q−1)2P (by (A1))
i i i i
= q2(q3P T P −q(q−q−1)2P T P −q(q−q−1)P
i i i i i i i
−2(q−q−1)P T P +(q−q−1)2P ) (by (E4∗))
i i i i
= q2((q3−q(q−q−1)2−2(q−q−1))P T P +(q−q−1)(−q+q−q−1)P )
i i i i
= q(P T P −(q−q−1)P )
i i i i
= P .
i+1
Finally, we prove (E4). First let j <i. Then, by (E4∗),
P T =q(P T P T −(q−q−1)P T ) =q(qP T P −(q−q−1)qP ) =qP ,
i+1 j i i i j i j i i i i i+1
8 tom halverson and arun ram
and T P = qP is similar. Now we consider the case where j = i,
j i+1 i+1
P T = q(P T P T −(q−q−1)P T ) (by (E5))
i+1 i i i i i i i
= q(P T qP T P T −q(q−q−1)P T P T −(q−q−1)P T ) (by (E5))
i i i−1 i−1 i−1 i i i i−1 i i i
= q(qP P T T T P −q(q−q−1)P P T2−(q−q−1)P T ) (by (E1))
i i−1 i i−1 i i−1 i i−1 i i i
= q(qP T T T P −q(q−q−1)P T2−(q−q−1)P T ) (by (E2))
i i i−1 i i−1 i i i i
= q(qP T T T P −q(q−q−1)P T2 −(q−q−1)P T ) (by (A2))
i i−1 i i−1 i−1 i i i i
= q(q2P T T P −q(q−q−1)P T2−(q−q−1)P T ) (by (E4∗))
i i i−1 i−1 i i i i
= q(q2P T T P −q(q−q−1)2P T −q(q−q−1)P −(q−q−1)P T ) (by (A1))
i i i−1 i−1 i i i i i
= q(q2P T T P −q2(q−q−1)P T −q(q−q−1)P )
i i i−1 i−1 i i i
= q(q2P P T T P −q2(q−q−1)P T −q(q−q−1)P ) (by (E2))
i i−1 i i−1 i−1 i i i
= q(q2P T P T P −q2(q−q−1)P T −q(q−q−1)P ) (by (E1))
i i i−1 i−1 i−1 i i i
= q(qP T P +q2(q−q−1)P T P −q2(q−q−1)P T −q(q−q−1)P ) (by (E5))
i i i i i i−1 i i i
= q(P +q2(q−q−1)P T P −q2(q−q−1)P T ) (by (E5))
i+1 i i i−1 i i
= q(P +q2(q−q−1)P P T −q2(q−q−1)P T ) (by (E1))
i+1 i i−1 i i i
= q(P +q2(q−q−1)P T −q2(q−q−1)P T ) (by (E2))
i+1 i i i i
= qP .
i+1
The case T P =qP is similar.
i i+1 i+1
Propositions 1.5 and 1.9 give the following theorem.
Theorem 1.10. R (q)∼= A (q) and thus (1.6) is a new presentation of R (q).
k k k
2. Hecke algebras
The affine Hecke algebra H˜
k
Fix q ∈ C∗. The affine Hecke algebra H˜ is the algebra given by generators
k
X ,...,X and T ,...,T
1 k 1 k−1
with relations
(1) T T =T T , |i−j| >1,
i j j i
(2) T T T =T T T , 1 ≤i ≤k−2,
i i+1 i i+1 i i+1
(3) T2 =(q−q−1)T +1, 1 ≤i ≤k,
i i
(4) X X = X X , 1 ≤i,j ≤ k,
i j j i
(5) X T =T X +(q−q−1)X , 1 ≤i ≤k−1.
i i i i+1 i
It follows from relations (3) and (5) that
X =T ···T T X T T ···T , for 1 ≤ i ≤k,
i i−1 2 1 1 1 2 i−1
rook monoid algebras and hecke algebras 9
and from (4) that
(6) X T X T = T X T X .
1 1 1 1 1 1 1 1
In fact, H˜ can be presented as the algebra generated by X and T ,...,T with relations (1-3)
k 1 1 k−1
and (6).
Let
[k]! = [1][2]···[k], where [i] =1+q2 +···+q2(i−1).
When [k]! 6= 0 a large class of irreducible representations of the affine Hecke algebra H˜ (q), the
k
integrallycalibratedirreduciblerepresentations,have a simple combinatorialconstruction. An H˜ -
k
module M is integrally calibrated if M has a basis of simultaneous eigenvectors for X ,X ,...,X
1 2 k
forwhichtheeigenvaluesarealloftheformqj withj ∈ Z. TheconstructionoftheseH˜ -modulesis
k
originally due to Cherednik [Ch] (see [Ra] for greaterdetail) and is a generalization of the classical
seminormal construction of the irreducible representations of the symmetric group by A. Young.
Young’sconstructionhadbeengeneralizedtoIwahori-Heckealgebrasofclassicaltype(seeTheorem
2.8 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)= 5 and |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. A standard 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 is a box in λ/µ define
CT(b) =q2(c−r), if b is in position (r,c) of λ. (2.1)
Theorem 2.2. ([Ch], see also [Ra, Theorem4.1], and[OR, Theorem6.20a]) Assume that[k]! 6=0.
Then the calibrated irreducible representations Hλ/µ of the affine Hecke algebra H˜ are indexed
k
by skew shapes and can be given explicitly as the vector space
Hλ/µ = C-span{v | L is a standard tableau of shape λ/µ}
L
(so that the symbols v form a basis of Hλ/µ) with H˜ -action given by
L k
X v = CT(L(i))v ,
i L L
CT(L(i+1))(q−q−1) CT(L(i+1))(q−q−1)
T v = v + q−1 + v ,
i L CT(L(i+1))−CT(L(i)) L CT(L(i+1))−CT(L(i)) siL
(cid:18) (cid:19) (cid:18) (cid:19)
where
s L is the same as L except i and i+1 are switched, and
i
10 tom halverson and arun ram
v =0, if s L is not a standard tableau.
siL i
Remark 2.3. In [OR] it is explained how the basis v of Hλ/µ and the action of H˜ in Theorem
L k
2.2 can be derived in a natural way from the general mechanism of quantum groups (R-matrices,
quantum Casimirs, the tensor product rule in (3.1)) and a Schur-Weyl duality theorem (Theorem
3.3 below) for the affine Hecke algebra.
The cyclotomic Hecke algebra H (u ,...,u ;q)
k 1 r
Let u ,...,u ∈ C and q ∈ C∗. The cyclotomic Hecke algebra H (u ,...,u ;q) is the quotient
1 r k 1 r
of the affine Hecke algebra H˜ by the ideal generated by the relation
k
(X −u )(X −u )···(X −u ) =0. (2.4)
1 1 1 2 1 r
The algebraH (u ,...,u ;q) is a deformationof the groupalgebraof the complexreflectiongroup
k 1 r
G(r,1,k) = (Z/rZ)≀S and is of dimension dim(H (u ,...,u ;q)) = rkk!. These algebras were
k k 1 r
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 2.5. ([Ar]) The algebra H (u ,,...,u ;q) is semisimple if and only if
k 1 r
q2du 6=u for all −k <d <k, 1 ≤i <j ≤ r, and [k]! 6=0,
i j
where [k]! =[1][2]···[k] and [i] =1+q2+···+q2(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 1.2. If T˜ = qT then T˜2 = q2((q − q−1)T + 1) =
i i i i
(q2−1)qT +q2 = (q2−1)T˜ +q2, which shows that our algebra is the same as Ariki’s except with
i i
parameter q2.
Ariki and Koike give a combinatorial construction of the irreducible representations of the
cyclotomic Hecke algebra H (u ,...,u ;q) when it is semisimple. Define
k 1 r
Hˆ(r) = {r-tuples λ =(λ(1),...,λ(r)) of partitions with k boxes total}. (2.6)
k
Let λ ∈ Hˆ(r). A standard tableau of shape λ is a filling of the boxes of λ with 1,2,...,k such that
k
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 q2(c−r), if the box b is in position (r,c) of λ(i). (2.7)
i
Theorem 2.8. [AK, Theorem 3.7] If H (u ,u ,...,u ;q) is semisimple its irreducible represen-
k 1 2 r
tations Hλ, λ ∈ Hˆ(r), are given by
k
Hλ = H(λ(1),...,λ(r)) = C-span{v | L is a standard tableau of shape λ}
L
