arXiv subjects

Arun Ram

Publications and source records attributed to Arun Ram.

At least 37 records · Page 2Linked to original sources

arXiv2018

Two boundary Hecke Algebras and combinatorics of type C

This paper gives a Schur-Weyl duality approach to the representation theory of the affine Hecke algebras of type C with unequal parameters. The first step is to realize the affine braid group of type $C_k$ as the group of braids on $k$ strands with two poles. Generalizing familiar methods from the one pole (type A) case, this provides commuting actions of the quantum group $U_q\mathfrak{g}$ and the affine braid group of type $C_k$ on a tensor space $M\otimes N \otimes V^{\otimes k}$. Special cases provide Schur-Weyl pairings between the affine Hecke algebra of type $C_k$ and the quantum group of type $\mathfrak{gl}_n$, resulting in natural labelings of many representations of the affine Hecke algebras of type C by partitions. Following an analysis of the structure of weights of affine Hecke algebra representations (extending the one parameter case to the three parameter case necessary for affine Hecke algebras of type C), we provide an explicit identification of the affine Hecke algebra representations that appear in tensor space (essentially by identifying their Langlands parameters).

math.RT↗

arXiv2018

The Steinberg-Lusztig tensor product theorem, Casselman-Shalika and LLT polynomials

In this paper we establish a Steinberg-Lusztig tensor product theorem for abstract Fock space. This is a generalization of the type A result of Leclerc-Thibon and a Grothendieck group version of the Steinberg-Lusztig tensor product theorem for representations of quantum groups at roots of unity. Although the statement can be phrased in terms of parabolic affine Kazhdan-Lusztig polynomials and thus has geometric content, our proof is combinatorial, using the theory of crystals (Littelmann paths). We derive the Casselman-Shalika formula as a consequence of the Steinberg-Lusztig tensor product theorem for abstract Fock space.

math.RT↗

arXiv2013

Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.

math.CO↗

arXiv2013

Symmetry breaking, subgroup embeddings and the Weyl group

We present a systematic approach to finding Higgs vacuum expectation values, which break a symmetry G to differently embedded isomorphic copies of a subgroup $H \subset G$ . We give an explicit formula for recovering each point in the vacuum manifold of a Higgs field which breaks G -> H. In particular we systematically identify the vacuum manifold G/H with linear combinations of the vacuum expectation values breaking G -> H_1 -> ... -> H_l. We focus on the most applicable case for current work on grand unified theories in extra dimensional models and low-energy effective theories for quantum chromodynamics. Here the subgroup, H, stabilizes the highest weight of the fundamental representation leading to a simple expression for each element of the vacuum manifold; especially for an adjoint Higgs field. These results are illustrated explicitly for adjoint Higgs fields and clearly linked to the mathematical formalism of Weyl groups. We use the final section to explicitly demonstrate how our work contributes to two contemporary high-energy physics research areas.

hep-th↗

arXiv2012

Generalized Schubert Calculus

In this paper we study the T-equivariant generalized cohomology of flag varieties using two models, the Borel model and the moment graph model. We study the differences between the Schubert classes and the Bott-Samelson classes. After setup of the general framework we compute, for classes of Schubert varieties of complex dimension <4 in rank 2 (including A_2, B_2, G_2 and A_1^{(1)}), moment graph representatives, Pieri-Chevalley formulas and products of Schubert classes. These computations generalize the computations in equivariant K-theory for rank 2 cases which are given in Griffeth-Ram arXiv:math/0405333.

math.RT↗

arXiv2012

Affine and degenerate affine BMW algebras: Actions on tensor space

The affine and degenerate affine Birman-Murakami-Wenzl (BMW) algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper we explain how the affine and degenerate affine BMW algebras are tantalizers (tensor power centralizer algebras) by defining actions of the affine braid group and the degenerate affine braid algebra on tensor space and showing that, in important cases, these actions induce actions of the affine and degenerate affine BMW algebras. We then exploit the connection to quantum groups and Lie algebras to determine universal parameters for the affine and degenerate affine BMW algebras. Finally, we show that the universal parameters are central elements--the higher Casimir elements for orthogonal and symplectic enveloping algebras and quantum groups.

math.RT↗

arXiv2011

Affine and degenerate affine BMW algebras: The center

The degenerate affine and affine BMW algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic Lie algebras and quantum groups, respectively. Cyclotomic BMW algebras, affine Hecke algebras, cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper the theory is unified by treating the orthogonal and symplectic cases simultaneously; we make an exact parallel between the degenerate affine and affine cases via a new algebra which takes the role of the affine braid group for the degenerate setting. A main result of this paper is an identification of the centers of the affine and degenerate affine BMW algebras in terms of rings of symmetric functions which satisfy a "cancellation property" or "wheel condition" (in the degenerate case, a reformulation of a result of Nazarov). Miraculously, these same rings also arise in Schubert calculus, as the cohomology and K-theory of isotropic Grassmanians and symplectic loop Grassmanians. We also establish new intertwiner-like identities which, when projected to the center, produce the recursions for central elements given previously by Nazarov for degenerate affine BMW algebras, and by Beliakova-Blanchet for affine BMW algebras.

math.RT↗

arXiv2011

Universal Specht modules for cyclotomic Hecke algebras

The graded Specht module $S^λ$ for a cyclotomic Hecke algebra comes with a distinguished generating vector $z^λ\in S^λ$, which can be thought of as a "highest weight vector of weight $λ$". This paper describes the {\em defining relations} for the Specht module $S^λ$ as a graded module generated by $z^λ$. The first three relations say precisely what it means for $z^λ$ to be a highest weight vector of weight $λ$. The remaining relations are homogeneous analogues of the classical {\em Garnir relations}. The homogeneous Garnir relations, which are {\em simpler} than the classical ones, are associated with a remarkable family of homogeneous operators on the Specht module which satisfy the braid relations.

math.RT↗

arXiv2011

Universal Verma modules and the Misra-Miwa Fock space

The Misra-Miwa $v$-deformed Fock space is a representation of the quantized affine algebra of type A. It has a standard basis indexed by partitions and the non-zero matrix entries of the action of the Chevalley generators with respect to this basis are powers of $v$. Partitions also index the polynomial Weyl modules for the quantum group $U_q(gl_N)$ as $N$ tends to infinity. We explain how the powers of $v$ which appear in the Misra-Miwa Fock space also appear naturally in the context of Weyl modules. The main tool we use is the Shapovalov determinant for a universal Verma module

math.QA↗

arXiv2010

A probabilistic interpretation of the Macdonald polynomials

The two-parameter Macdonald polynomials are a central object of algebraic combinatorics and representation theory. We give a Markov chain on partitions of k with eigenfunctions the coefficients of the Macdonald polynomials when expanded in the power sum polynomials. The Markov chain has stationary distribution a new two-parameter family of measures on partitions, the inverse of the Macdonald weight (rescaled). The uniform distribution on permutations and the Ewens sampling formula are special cases. The Markov chain is a version of the auxiliary variables algorithm of statistical physics. Properties of the Macdonald polynomials allow a sharp analysis of the running time. In natural cases, a bounded number of steps suffice for arbitrarily large k.

math.PR↗

arXiv2009

Representations of Khovanov-Lauda-Rouquier Algebras and Combinatorics of Lyndon Words

We construct irreducible representations of affine Khovanov-Lauda-Rouquier algebras of arbitrary finite type. The irreducible representations arise as simple heads of appropriate induced modules, and thus our construction is similar to that of Bernstein and Zelevinsky for affine Hecke algebras of type A. The highest weights of irreducible modules are given by the so-called good words, and the highest weights of the 'cuspidal modules' are given by the good Lyndon words. In a sense, this has been predicted by Leclerc.

math.RT↗

arXiv2008

Homogeneous Representations of Khovanov-Lauda Algebras

We construct irreducible graded representations of simply laced Khovanov-Lauda algebras which are concentrated in one degree. The underlying combinatorics of skew shapes and standard tableaux corresponding to arbitrary simply laced types has been developed previously by Peterson, Proctor and Stembridge. In particular, the Peterson-Proctor hook formula gives dimensions of the homogeneous irreducible modules corresponding to straight shapes.

math.RT↗

arXiv2008

Alcove walks, buildings, symmetric functions and representations

For a complex simple Lie algebra, the dimension $K_{λμ}$ of the $μ$ weight space of a finite dimensional representation of highest weight $λ$ is the same as the number of Littelmann paths of type $λ$ and weight $μ$. In this paper we give an explicit construction of a path of type $λ$ and weight $μ$ whenever $K_{λμ}\ne 0$. This construction has additional consequences, it produces an explicit point in the building which chamber retracts to $λ$ and sector retracts to $μ$, and an explicit point of the affine Grassmannian in the corresponding Mirković-Vilonen intersection. In an appendix we discuss the connection between retractions in buildings and alcove walks.

math.RT↗

arXiv2008

A combinatorial formula for Macdonald polynomials

In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for Macdonald polynomials for all Lie types. These formulas are generalizations of the formulas of Haglund-Haiman-Loehr for Macdonald polynoimals of type GL(n). At q=0 these formulas specialize to the formula of Schwer for the Macdonald spherical function in terms of positively folded alcove walks and at q=t=0 these formulas specialize to the formula for the Weyl character in terms of the Littelmann path model (in the positively folded gallery form of Gaussent-Littelmann).

math.CO↗

arXiv2008

Combinatorics in affine flag varieties

The Littelmann path moel gives a realization of the crystals of integrable representations of symmetrizable Kac-Moody Lie algebras. Recent work of Gaussent-Littelmann and others has demonstrated a connection between this model and the geometry of the loop Grassmannian. The alcove walk model is a version of the path model which is intimately connected to the combinatorics of the affine Hecke algebra. In this paper we define a refined alcove walk model which encodes the points of the affine flag variety. We show that this combinatorial indexing naturally indexes the "cells" in generalized Mirkovic-Vilonen intersections.

math.RT↗

arXiv2007

Commuting Families in Temperley-Lieb Algebras

We define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements in the affine Hecke algebra of type A, which in turn come from the Casimir element of the quantum group $U_h\mathfrak{gl}_n$. We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.

math.RT↗

arXiv2006

Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux

This paper makes precise the close connection between the affine Hecke algebra, the path model, and the theory of crystals. Section 2 is a basic pictorial exposition of Weyl groups and affine Weyl groups and Section 5 is an exposition of the theory of (a) symmetric functions, (b) crystals and (c) the path model. Sections 3 and 4 give an exposition of the affine Hecke algebra and recent results regarding the combinatorics of spherical functions on p-adic groups (Hall-Littlewood polynomials). The $q$-analogue of the theory of crystals developed in Section 4 specializes to the path model version of the ``classical'' crystal theory. The connection to the affine Hecke algebra and the approach to spherical functions for a $p$-adic group in Nelsen-Ram was made concrete by C. Schwer who told me that ``the periodic Hecke module encodes the positively folded galleries'' of Gaussent-Littelmann. This paper is a further development of this point of view.

math.RT↗

arXiv2005

The quantum black hole

The quantum nature of a black hole is revealed using the simplest terms that one learns in undergraduate and beginning graduate courses. The exposition demonstrates -- vividly -- the importance and power of the quantum oscillator in contemporary research in theoretical physics.

gr-qc↗