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School of Mathematics and Statistics - Research Publications
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ItemNo Preview AvailableTHE STEINBERG-LUSZTIG TENSOR PRODUCT THEOREM, CASSELMAN-SHALIKA, AND LLT POLYNOMIALSLanini, M ; Ram, A (AMER MATHEMATICAL SOC, 2019-04-02)
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.
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ItemHopf algebras and Markov chains: two examples and a theoryDiaconis, P ; Pang, CYA ; Ram, A (SPRINGER, 2014-05)
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ItemAFFINE AND DEGENERATE AFFINE BMW ALGEBRAS: THE CENTERDaugherty, Z ; Ram, A ; Virk, R (OSAKA JOURNAL OF MATHEMATICS, 2014-01)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.
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ItemAffine Braids, Markov traces and the category OOrellana, ; RAM, A (American Mathematical Society, 2007)
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ItemCOMMUTING FAMILIES IN HECKE AND TEMPERLEY-LIEB ALGEBRASHalverson, T ; Mazzocco, M ; Ram, A (NAGOYA UNIV, 2009-09)Abstract 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 . We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.
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ItemHomogeneous representations of Khovanov-Lauda algebrasKleshchev, A ; Ram, A (EUROPEAN MATHEMATICAL SOC, 2010)
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ItemCombinatorics in affine flag varietiesParkinson, J ; Ram, A ; Schwer, C (ACADEMIC PRESS INC ELSEVIER SCIENCE, 2009-06-01)