Last update: 3 March 2013
In this paper all of the classical constructions of A. Young are generalized to affine Hecke algebras of type A. It is proved that the calibrated irreducible representations of the affine Hecke algebra are indexed by placed skew shapes and that these representations can be constructed explicitly with a generalization of Young’s seminormal construction of the irreducible representations of the symmetric group. The seminormal construction of an irreducible calibrated module does not produce a basis on which the affine Hecke algebra acts integrally but using it one is able to pick out a different basis, an analogue of Young’s natural basis, which does generate an integral lattice in the module. Analogues of the “Garnir relations” play an important role in the proof. The Littlewood-Richardson coefficients arise naturally as the decomposition multiplicities for the restriction of an irreducible representation of the affine Hecke algebra to the Iwahori-Hecke algebra.
My recent work [Ram1998], [Ram1998-2] on the representations of affine Hecke algebras has been strongly motivated by the classical theory of Young tableaux. This research has resulted in the generalization of many of A. Young’s constructions to general finite root systems. With these generalizations of standard Young tableaux one is able to use Young’s classical “seminormal construction” to construct irreducible representations of affine Hecke algebras corresponding to arbitrary finite crystallographic root systems.
Because the classical combinatorics of Young tableaux is much more advanced than that of the newly developed generalization it is often possible to give simpler proofs and more extensive results for the case of affine Hecke algebras of type A. The purpose of this paper is to compile some of these results and proofs. In particular, we obtain a generalization of Young’s natural basis and derive certain induction and restriction rules which are not yet available in the general case. It will also be more clear from the exposition here, how the generalization of the Young tableau theory given in [Ram1998] and [Ram1998-2] relates to the classical setup, something which is not always obvious when working in the general root system context.
The main results
These representations are indexed by placed skew shapes. The dimension of an irreducible calibrated representation is the number of standard tableaux of the corresponding skew shape and the representation is constructed by explicit formulas which give the action of each generator of the affine Hecke algebra on a specific basis, the elements of which are indexed by standard Young tableaux. This is a generalization of the constructions of A. Young [You1931,You1934], P. Hoefsmit [Hoe1974], H. Wenzl [Wen1988], and Ariki and Koike [AKo1994] (see [Ram1997] for a review of some of the unpublished results of Hoefsmit). Parts of Theorem 4.1 were first discovered by Cherednik and are stated (without proof) in [Che1987]. I am grateful to A. Zelevinsky for pointing this out to me and to I. Cherednik for some informative discussions.
Young’s natural basis is the one that is most often used in the study of irreducible representations of the symmetric group, it is the one that is usually taken as the basis of the “Specht module”, see for example [JKe1981], [Sag1991], [Ful1997]. It has the wonderful property that it is an integral basis for the module, i.e. the matrices representating the action of the symmetric group on this basis contain integer entries. This is especially important because it opens the door to a combinatorial study of the modular representations of the symmetric group.
Using the analogue of the seminormal basis for the irreducible calibrated representations of the affine Hecke algebra we can define an analogue of Young’s natural basis in each of these representations. As desired, this basis is an integral basis of the module; the matrices representing the action of the affine Hecke algebra on this basis have all entries in the ring $\mathbb{Z}[q,{q}^{-1}]\text{.}$ These results are a $q\text{-analogue}$ of some of the results in [GWa1989].
One of the pleasant surprises one has when generalizing Young’s natural basis from this point of view is that the “Garnir relations” take a particularly simple form: If $\left\{{v}_{L}\right\}$ is Young’s seminormal basis and $\left\{{n}_{L}\right\}$ is Young’s natural basis then the relations
$${v}_{L}=0,\phantom{\rule{2em}{0ex}}\text{when}\hspace{0.17em}L\hspace{0.17em}\text{is not a standard tableau,}$$are the Garnir relations. One recovers the Garnir relations in their classical form by expanding the ${v}_{L}$ in terms of the ${n}_{L}\text{.}$
This result gives a completely new (and unexpected) representation theoretic interpretation of the Littlewood-Richardson coefficients.
Until now skew shapes have appeared in the combinatorial literature as something of a novelty, a useful combinatorial tool which indexes some strangely well-behaved representations of the symmetric group. It has always been a surprise that the combinatorics of the irreducible representations of the symmetric group generalizes so beautifully to this special class of highly reducible representations of the symmetric group.
This fact is no longer strange. In fact, these representations are irreducible representations of the affine Hecke algebra, and thus are basic and fundamental. Several of the skew Schur function identities in [Mac1995] I can be given representation theoretic interpretations in this context, see Theorem 6.2 and Corollary 6.3.
This paper is part of a series [Ram1998,Ram1998-2,Ram1998-3] [RRa1998,RRa1998-2] of papers on representations of affine Hecke algebras. During this work I have benefited from conversations with many people. To choose only a few, there were discussions with S. Fomin, F. Knop, L. Solomon, M. Vazirani and N. Wallach which played an important role in my progress. There were several times when I tapped into J. Stembridge’s fountain of useful knowledge about root systems. G. Benkart was a very patient listener on many occasions. H. Barcelo, P. Deligne, T. Halverson, R. Macpherson and R. Simion all gave large amounts of time to let me tell them my story and every one of these sessions was helpful to me in solidifying my understanding.
I single out Jacqui Ramagge with special thanks for everything she has done to help with this project: from the most mundane typing and picture drawing to deep intense mathematical conversations which helped to sort out many pieces of this theory. Her immense contribution is evident in that some of the papers in this series on representations of affine Hecke algebras are joint papers.
A portion of this research was done during a semester stay at Mathematical Sciences Research Institute where I was supported by a Postdoctoral Fellowship. I thank MSRI and National Science Foundation for support of my research.
This is an excerpt of the preprint entitled Skew shape representations are irreducible authored by Arun Ram in 1998.
Research supported in part by National Science Foundation grant DMS-9622985, and a Postdoctoral Fellowship at Mathematical Sciences Research Institute.