Searcharxiv⌕ Search

arXiv subjects

Arun Ram

Publications and source records attributed to Arun Ram.

At least 55 records · Page 3Linked to original sources
arXiv2004

Affine Hecke algebras and the Schubert calculus

Using a combinatorial approach which avoids geometry, this paper studies the ring structure of K_T(G/B), the T-equivariant K-theory of the (generalized) flag variety G/B. Here the data is a complex reductive algebraic group (or symmetrizable Kac-Moody group) G, a Borel subgroup B, and a maximal torus T, and K_T(G/B) is the Grothendieck group of T-equivariant coherent sheaves on G/B. We prove "Pieri-Chevalley" formulas for the products of a Schubert class by a homogeneous line bundle (dominant or anti-dominant) and for products of a Schubert class by a codimension 1 Schubert class. All of these Pieri-Chevalley formulas are given in terms of the combinatorics of the Littelmann path model. We give explicit computations of products of Schubert classes for the rank two cases and this data allows us to make a "positivity conjecture" generalizing the theorems of Brion and Graham, which treat the cases K(G/B) and H_T^*(G/B), respectively.

math.RT↗
arXiv2004

Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory

We show that the Young tableaux theory and constructions of the irreducible representations of the Weyl groups of type A, B and D, Iwahori-Hecke algebras of types A, B, and D, the complex reflection groups G(r,p,n) and the corresponding cyclotomic Hecke algebras H_{r,p,n}, can be obtained, in all cases, from the affine Hecke algebra of type A. The Young tableaux theory was extended to affine Hecke algebras (of general Lie type) in recent work of A. Ram. We also show how (in general Lie type) the representations of general affine Hecke algebras can be constructed from the representations of simply connected affine Hecke algebras by using an extended form of Clifford theory. This extension of Clifford theory is given in the Appendix.

math.RT↗
arXiv2004

Calibrated representations of affine Hecke algebras

This paper introduces the notion of calibrated representations for affine Hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations. The main results are that (1) irreducible calibrated representations are indexed by placed skew shapes, (2) the dimension of an irreducible calibrated representation is the number of standard Young tableaux corresponding to the placed skew shape and (3) each irreducible calibrated representation is constructed explicitly by formulas which describe the action of each generator of the affine Hecke algebra on a specific basis in the representation space. This construction is a generalization of A. Young's seminormal construction of the irreducible representations of the symmetric group. In this sense Young's construction has been generalized to arbitrary Lie type.

math.RT↗
arXiv2004

Standard Young tableaux for finite root systems

The study of representations of affine Hecke algebras has led to a new notion of shapes and standard Young tableaux which works for the root system of any finite Coxeter group. This paper is completely independent of affine Hecke algebra theory and is purely combinatorial. We define generalized shapes and standard Young tableaux and show that these new objects coincide with the classical ones for root systems of Type A. The classical notions of conjugation of shapes, ribbon shapes, axial distances, and the row reading and column reading standard tableaux, have natural generalizations to the root system case. In the final section we give an interpretation of the shapes and standard tableaux for root systems of Type C which is in a form similar to classical theory of shapes and standard tableaux.

math.RT↗
arXiv2004

Skew shape representations are irreducible

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.

math.RT↗
arXiv2004

A Pieri-Chevalley formula in the K-theory of a G/B-bundle

The main result of this announcement is a formula for the tensor product of the class of a homogeneous line bundle with a Schubert class, expressed as a K(X)-linear combination of Schubert classes. We believe that this formula is the most general uniform result in the intersection theory of Schubert classes: it is related to a recent result of Fulton and Lascoux who presented a similar formula for a GL_n(C)/B-bundle. Indeed, in this case, their formula and ours coincide once one knows how to translate between their combinatorics with tableaux and ours with Littelmann paths. O. Mathieu has also proved the positivity which is implied by our formula. Applying the Chern character to our formula, and equating the lowest order terms we obtain a relative version of the classical result of Chevalley alluded to in the title of this paper.

math.RT↗
arXiv2004

A Pieri-Chevalley formula for K(G/B)

The ring K(G/B) is isomorphic to a quotient of a Laurent polynomial ring by an ideal generated by certain W-symmetric functions and has a basis given by classes O_w, where O_w is the class of the structure sheaf of the Schubert variety X_w. In this paper we give an explicit combinatorial formula for the tensor product of a (negative) line bundle with the structure sheaf of a Schubert variety, expanded in terms of the Schubert class basis O_w. Chevalley's classical formula for the class of a hyperplane section of a Schubert variety in H^*(G/B) can be recovered from our formula by applying the Chern character and comparing lowest degree terms. The higher order terms of our formula may yield further interesting identities in cohomology. Fulton and Lascoux have given a formula similar to ours for the GL_n(C) case. Our formula is a generalization of their formula to general type. In our work the column strict tableaux used by Fulton and Lascoux are replaced by Littelmann's path model.

math.RT↗
arXiv2004

Partition Algebras

The partition algebras are algebras of diagrams (which contain the group algebra of the symmetric group and the Brauer algebra) such that the multiplication is given by a combinatorial rule and such that the structure constants of the algebra depend polynomially on a parameter. This is a survey paper which proves the primary results in the theory of partition algebras. Some of the results in this paper are new. This paper gives: (a) a presentation of the partition algebras by generators and relations, (b) shows that each partition algebra has an ideal which is isomorphic to a basic construction and such that the quotient is isomorphic to the group algebra of the symmetric gropup, (c) shows that partition algebras are in "Schur-Weyl duality" with the symmetric groups on tensor space, (d) provides a construction of "Specht modules" for the partition algebras (integral lattices in the generic irreducible modules), (e) determines (with a couple of exceptions) the values of the parameter where the partition algebras are semisimple, (f) provides "Murphy elements" for the partition algebras that play exactly analogous roles to the classical Murphy elements for the group algebra of the symmetric group. The primary new results in this paper are (a) and (f).

math.RT↗
arXiv2004

Affine braids, Markov traces and the category O

This paper provides a unified approach to results on representations of affine Hecke algebras, cyclotomic Hecke algebras, affine BMW algebras, cyclotomic BMW algebras, Markov traces, Jacobi-Trudi type identities, dual pairs (Zelevinsky), and link invariants (Turaev). The key observation in the genesis of this paper was that the technical tools used to obtain the results in Orellana and Suzuki, two a priori unrelated papers, are really the same. Here we develop this method and explain how to apply it to obtain results similar to those in Orellana and Suzuki in more general settings. Some specific new results which are obtained are the following: (a) a generalization of the results on Markov traces obtained by Orellana to centralizer algebras coming from quantum groups of all Lie types, (b) a generalization of the results of Suzuki to show that Kazhdan-Lusztig polynomials of all finite Weyl groups occur as decomposition numbers in the representation theory of affine braid groups of type A, (c) a generalization of the functors used by Zelevinsky to representations of affine braid groups of type A, (d) a definition of the affine BMW-algebra (Birman-Murakami-Wenzl) and show that it has a representation theory analogous to that of affine Hecke algebras. In particular there are ``standard modules'' for these algebras which have composition series where multiplicites of the factors are given by Kazhdan-Lusztig polynomials for Weyl groups of types A,B,and C, (e) we generalize the results of Leduc and Ram on constructing representations of centralizer algebras to affine centralizer algebras.

math.RT↗
arXiv2004

Representations of rank two affine Hecke algebras

This paper classifies and constructs explicitly all the irreducible representations of affine Hecke algebras of rank two root systems. The methods used to obtain this classification are primarily combinatorial and are, for the most part, an application of the methods used in a recent preprint of Ram. A special effort has been made special effort to describe how the classification here relates to the classifications by Langlands parameters (coming from p-adic group theory) and by indexing triples (coming from a q-analogue of the Springer correspondence). The examples here illustrate and clarify results of Ram, Kazhdan-Lusztig, Chriss-Ginzburg, Barbasch-Moy, Evens, Kriloff, and Heckman-Opdam. Much of the power of the combinatorial methods which are now available is evident from the calculations in this paper. One hopes that eventually there will be a combinatorial construction of all irreducible representations of all affine Hecke algebras. Such a construction may depend heavily on the rank two cases. This idea is analogous to the way that the rank two cases are the basic building blocks in the presentations of Coxeter groups by ``braid'' relations and the presentations of Kac-Moody Lie algebras (and quantum groups) by Serre relations.

math.RT↗
arXiv2004

Affine Hecke algebras and generalized standard Young tableaux

This paper introduces calibrated representations for affine Hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations. The primary technique is to provide indexing sets for controlling the weight space structure of finite dimensional modules for the affine Hecke algebra. Using these indexing sets we show that (1) irreducible calibrated representations are indexed by skew local regions, (2) the dimension of an irreducible calibrated representation is the number of chambers in the local region, (3) each irreducible calibrated representation is constructed explicitly by formulas which describe the action of the generators of the affine Hecke algebra on a specific basis in the representation space. The indexing sets for weight spaces are generalizations of standard Young tableaux and the construction of the irreducible calibrated affine Hecke algebra modules is a generalization of A. Young's seminormal construction of the irreducible representations of the symmetric group. In this sense Young's construction has been generalized to arbitrary Lie type.

math.RT↗
arXiv2004

Kostka-Foulkes polynomials and Macdonald spherical functions

Generalized Hall-Littlewood polynomials (Macdonald spherical functions) and generalized Kostka-Foulkes polynomials ($q$-weight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics. This paper attempts to organize the different definitions of these objects and prove the fundamental combinatorial results from ``scratch'', in a presentation which, hopefully, will be accessible and useful for both the nonexpert and researchers currently working in this very active field. The combinatorics of the affine Hecke algebra plays a central role. The final section of this paper can be read independently of the rest of the paper. It presents, with proof, Lascoux and Schützenberger's positive formula for the Kostka-Foulkes poynomials in the type A case.

math.RT↗
arXiv2002

Classification of graded Hecke algebras for complex reflection groups

The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V). This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction. The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups.

math.GR↗
arXiv1997

Combinatorial Representation Theory

We attempt to survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. We give a personal viewpoint on the field, while remaining aware that there is much important and beautiful work that we have not been able to mention.

math.RT↗
arXiv1997

Explicit irreducible representations of the Iwahori-Hecke algebra of type $F_4$

A general method for computing irreducible representations of Weyl groups and Iwahori-Hecke algebras was introduced earlier by the first author. The representations of the algebras of types $A_n$, $B_n$, $D_n$ and $G_2$ were computed and it is the purpose of this paper to extend these computations to $F_4$. The main goal here is to compute irreducible representations of the Iwahori-Hecke algebra of type $F_4$ by only using information in the character table of the Weyl group.

math.RT↗
arXiv1996

Robinson-Schensted-Knuth insertion and characters of symmetric groups and Iwahori-Hecke algebras of type A

The purpose of this note is to give an insertion scheme proof of the formula, $$p_μ= \sum_{λ\vdash k} χ^λ(μ)s_λ,\formula$$ where $p_μ$ is the power sum symmetric function, $s_λ$ is the Schur function and $χ^λ(μ)$ is the irreducible character of the symmetric group $S_k$ indexed by the partition $λ$ and evaluated at a permutation of cycle type $μ=(μ_1,\ldots,μ_\ell)$. The proof of this formula is by direct application of the Robinson-Schensted-Knuth insertion scheme and a recent formula of Roichman.

math.RT↗