Advanced Studies in Pure Mathematics

Rigged Configurations and Catalan, stretched parabolic Kostka numbers and polynomials: Polynomiality, unimodality and log-concavity

Anatol N. Kirillov

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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.

Article information

Source
Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, H. Konno, H. Sakai, J. Shiraishi, T. Suzuki and Y. Yamada, eds. (Tokyo: Mathematical Society of Japan, 2018), 303-346

Dates
Received: 18 August 2015
Revised: 15 December 2015
First available in Project Euclid: 21 September 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1537499430

Digital Object Identifier
doi:10.2969/aspm/07610303

Mathematical Reviews number (MathSciNet)
MR3837926

Zentralblatt MATH identifier
07039307

Subjects
Primary: 05E05: Symmetric functions and generalizations 05E10: Combinatorial aspects of representation theory [See also 20C30] 05A19: Combinatorial identities, bijective combinatorics

Keywords
Catalan and Narayana numbers stretched Kostka polynomials internal product of Schur functions rigged configurations

Citation

Kirillov, Anatol N. Rigged Configurations and Catalan, stretched parabolic Kostka numbers and polynomials: Polynomiality, unimodality and log-concavity. Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, 303--346, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07610303. https://projecteuclid.org/euclid.aspm/1537499430


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