Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 17 September 2013

Appendix B

B5. The Iwahori-Hecke algebras Hk(q) of type A

A k-braid is viewed as two rows of k vertices, one above the other, and k strands that connect top vertices to bottom vertices in such a way that each vertex is incident to precisely one strand. Strands cross over and under each other in three-space as they pass from one vertex to the next. t1= ,t2= . We multiply k-braids t1 and t2 using the concatenation product given by identifying the vertices in the top row of t2 with the corresponding vertices in the bottom row of t1 to obtain the product t1t2. t1t2= ,

Given a permutation wSk we will make a k-braid Tw by tracing the edges in order from left to right across the top row. Any time an edge that we are tracing crosses an edge that has been already traced we raise the pen briefly so that the edge being traced goes under the edge which is already there. Applying this process to all of the permutations in Sk produces a set of k! braids. w= T2=

Fix q. The Iwahori-Hecke algebra Hk(q) of type Ak-1 is the span of the k! braids produced by tracing permutations in Sk with multiplication determined by the braid multiplication and the following identity. =(q-1) +q . This identity can be applied in any local portion of the braid.

Theorem B5.1. The algebra Hk(q) is the associative algebra over presented by generators T1,,Tk-1 and relations TiTj=TjTi, for|i-j|>1, TiTi+1Ti=Ti+1TiTi+1, for1in-2, Ti2=(q-1)Ti+q, for2in.

The Iwahori-Hecke algebra of type A is a q-analogue of the group algebra of the symmetric group. If we allow ourselves to be imprecise (about the limit) we can write limq1Hk (q)=Sk.

Let q be a power of a prime G=GL(n,𝔽q) where Fq is the finite field with q elements. Let B be the subgroup of upper triangular matrices in G and let 1BG be the trivial representation of B induced to G, i.e. the G-module given by 1BG=-span {gB|gG}, where G acts on the cosets by left multiplication. Using the description, see §B3, of Hn(q) as a double coset algebra one gets an action of Hn(q) on 1BG, by right multiplication. This action commutes with the G action.

Theorem B5.2.

(a) The action of Hn(q) on 1BG generates EndG(1BG).
(b) The action of G on 1BG generates EndHn(q)(1BG).

This theorem gives a “duality” between GL(n,𝔽q) and Hn(q) which is similar to a Schur-Weyl duality, but it differs in a crucial way: the representation 1BG is not a tensor power representation, and thus this is not yet realizing Hn(q) as a tensor power centralizer.

The following result gives a true analogue of the Schur-Weyl duality for the Iwahori-Hecke algebra of type A, it realizes Hk(q) as a tensor power centralizer. Assume that q is not 0 and is not a root of unity. Let Uq𝔰𝔩n be the Drinfel’d-Jimbo quantum group of type An-1 and let V be the n-dimensional irreducible representation of Uq𝔰𝔩n with highest weight ω1. There is an action, see [CPr1994], of Hk(q2) on Vk which commutes with the Uq𝔰𝔩n action.

Theorem B5.3.

(a) The action of Hk(q2) on Vk generates EndUq𝔰𝔩n(Vk).
(b) The action of Uq𝔰𝔩n on Vk generates EndHk(q2)(Vk).

Theorem B5.4. The Iwahori-Hecke algebra of type Ak-1, Hk(q), is semisimple if and only if q0 and q is not a jth root of unity for any 2jn.

Partial results for Hk(q)

The following results giving answers to the main questions (Ia-c) for the Iwahori-Hecke algebras of type A hold when q is such that Hk(q) is semisimple.

I. What are the irreducible Hk(q)-modules?

(a) How do we index/count them?
There is a bijection Partitionsλof n1-1 Irreducible representationsHλ
(b) What are their dimensions?
The dimension of the irreducible representation Hλ is given by dim(Hλ) = # of standard tableaux of shapeλ = n!xλhx,
(c) What are their characters?
For each partition μ=(μ1,μ2,,μ) of k let χλ(μ) be the character of the irreducible representation Hλ evaluated at the element Tγμ where γμ is the permutation γμ= μ1 μ2 μ .
Then the character χλ(μ) is given by χλ(μ)=T wtμ(T), where the sum is over all standard tableaux T of shape λ and wtμ(T)=i=1n f(i,T), where f(i,T)= { -1, ifiB(μ) andi+1 is sw ofi, 0, ifi,i+1 B(μ), i+1is ne ofi , andi+2 is sw ofi+1, q, otherwise, and B(μ)={μ1+μ2++μk|1k}. In the formula for f(i,T), sw means strictly south and weakly west and ne means strictly north and weakly east.


The book [CPr1994] contains a treatment of the Schur-Weyl duality type theorem given above. See also the references there. Several basic results on the Iwahori-Hecke algebra are given in the book [GHJ1989]. The theorem giving the explicit values of q such that Hk(q) is semisimple is due to Gyoja and Uno [GUn1989]. The character formula given above is due to Roichman [Roi1997]. See [Ram1997-2] for an elementary proof.

Notes and references

This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.

Key words and phrases. Algebraic combinatorics, representations.

Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..

page history