Translator Disclaimer
VOL. 76 | 2018 Rigged Configurations and Catalan, stretched parabolic Kostka numbers and polynomials: Polynomiality, unimodality and log-concavity
Anatol N. Kirillov

Editor(s) Hitoshi Konno, Hidetaka Sakai, Junichi Shiraishi, Takao Suzuki, Yasuhiko Yamada

Abstract

We will look at the Catalan numbers from the Rigged Configurations point of view originated [10] from an combinatorial analysis of the Bethe Ansatz Equations associated with the higher spin anisotropic Heisenberg models. Our strategy is to take a combinatorial interpretation of the Catalan number $C_n$ as the number of standard Young tableaux of rectangular shape $(n^2)$, or equivalently, as the Kostka number $K_{(n^2),1^{2n}}$, as the starting point of our research. We observe that the rectangular (or multidimensional) Catalan numbers $ C(m,n)$, introduced and studied by P. MacMahon [23], [34], see also [35], can be identified with the corresponding Kostka numbers $K_{(n^m),1^{mn}}$, and therefore can be treated by the Rigged Configurations technique. Based on this technique we study the stretched Kostka numbers and polynomials, and give a proof of a strong rationality of the stretched Kostka polynomials. This result implies a polynomiality property of the stretched Kostka and stretched Littlewood–Richardson coefficients [8], [28], [17]. Finally, we give a brief introduction to a rigged configuration version of the Robinson–Schensted–Knuth correspondence.

Another application of the Rigged Configuration technique presented, is a new family of counterexamples to Okounkov's log-concavity conjecture [27].

Finally, we apply Rigged Configurations technique to give a combinatorial proof of the unimodality of the principal specialization of the internal product of Schur functions. In fact we prove a combinatorial (fermionic) formula for generalized $q$-Gaussian polynomials which is a far generalization of the so-called $KOH$-identity [26], as well as it manifests the unimodality property of the $q$-Gaussian polynomials.

Information

Published: 1 January 2018
First available in Project Euclid: 21 September 2018

zbMATH: 07039307
MathSciNet: MR3837926

Digital Object Identifier: 10.2969/aspm/07610303

Subjects:
Primary: 05A19, 05E05, 05E10

Rights: Copyright © 2018 Mathematical Society of Japan

PROCEEDINGS ARTICLE
44 PAGES


SHARE
RIGHTS & PERMISSIONS
Get copyright permission
Back to Top