Cyclotomic and Littlewood Polynomials Associated to Algebras

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials, $k$-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space. In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$.

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A Combinatorial Approach to the $q,t$-Symmetry Relation in Macdonald Polynomials

The Electronic Journal of Combinatorics ◽

10.37236/5350 ◽

2016 ◽

Vol 23 (2) ◽

Cited By ~ 3

Author(s):

Maria Monks Gillespie

Keyword(s):

Macdonald Polynomials ◽

Combinatorial Proof ◽

Symmetry Relation ◽

Combinatorial Approach ◽

Combinatorial Formula ◽

Recursive Structure ◽

Littlewood Polynomials

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation $\widetilde{H}_\mu(\mathbf{x};q,t)=\widetilde{H}_{\mu^\ast}(\mathbf{x};t,q)$. We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials ($q=0$) when $\mu$ is a partition with at most three rows, and for the coefficients of the square-free monomials in $\mathbf{x}$ for all shapes $\mu$. We also provide a proof for the full relation in the case when $\mu$ is a hook shape, and for all shapes at the specialization $t=1$. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.

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A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

Canadian Journal of Mathematics ◽

10.4153/cjm-2011-078-4 ◽

2012 ◽

Vol 64 (4) ◽

pp. 822-844 ◽

Cited By ~ 20

Author(s):

J. Haglund ◽

J. Morse ◽

M. Zabrocki

Keyword(s):

Building Blocks ◽

Main Diagonal ◽

Combinatorial Theory ◽

Dyck Path ◽

Dyck Paths ◽

Littlewood Polynomials ◽

Diagonal Harmonics ◽

Littlewood Polynomial ◽

Theory Operator ◽

Shuffle Conjecture

Abstract We introduce a q, t-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory ∇ operator applied to a Hall–Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the “shuffle conjecture” (Duke J. Math. 126 (2005), pp. 195 − 232) for ∇ en[X]. We bring to light that certain generalized Hall–Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of q, t-Catalan sequences, and we prove a number of identities involving these functions.

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Limit theorems for isotropic random walks on triangle buildings

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700008995 ◽

2002 ◽

Vol 73 (3) ◽

pp. 301-334 ◽

Cited By ~ 3

Author(s):

Marc Lindlbauer ◽

Michael Voit

Keyword(s):

Limit Theorem ◽

Random Walks ◽

Local Limit Theorem ◽

Local Limit ◽

Two Dimensional ◽

Littlewood Polynomials ◽

Large Numbers ◽

Moment Functions ◽

Vertex Sets ◽

Isotropic Random

AbstractThe spherical functions of triangle buildings can be described in terms of certain two-dimensional orthogonal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynimial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups. We present strong laws of large numbers, a central limit theorem, and a local limit theorem; all these results are well-known for homogeneous trees. Proofs are based on moment functions on hypergroups and on explicit expansions of the hypergroup characters in terms of certain two-dimensional Tchebychev polynimials.

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