[Paper Review] A $q$-deformation of an algebra of Klyachko and Macdonald's reduced word formula

Philippe Nadeau, Vasu Tewari|arXiv (Cornell University)|Jun 7, 2021

Advanced Combinatorial Mathematics35 references4 citations

TL;DR

This paper introduces a q-deformation of an algebraic structure related to Klyachko's work and Macdonald's reduced word formula, establishing a q-analog of Eulerian numbers via Schur functions and hook-lengths. The key result expresses a q-generating function for certain tableaux in terms of q-hook-lengths and major index, generalizing classical Eulerian number identities to a q-deformed setting with combinatorial interpretation via standard Young tableaux.

ABSTRACT

There is a striking similarity between Macdonald's reduced word formula and the image of the Schubert class in the cohomology ring of the permutahedral variety $\mathrm{Perm}_n$ as computed by Klyachko. Toward understanding this better, we undertake an in-depth study of a $q$-deformation of the $\mathbb{S}_n$-invariant part of the rational cohomology ring of $\mathrm{Perm}_n$, which we call the $q$-Klyachko algebra. We uncover intimate links between expansions in the basis of squarefree monomials in this algebra and various notions in algebraic combinatorics, thereby connecting seemingly unrelated results by finding a common ground to study them. Our main results are as follows. 1) A $q$-analog of divided symmetrization ($q$-DS) using Yang-Baxter elements in the Hecke algebra. It is a linear form that picks up coefficients in the squarefree basis. 2) A relation between $q$-DS and the ideal of quasisymmetric polynomials involving work of Aval--Bergeron--Bergeron. 3) A family of polynomials in $q$ with nonnegative integral coefficients that specialize to Postnikov's mixed Eulerian numbers when $q=1$. We refer to these new polynomials as remixed Eulerian numbers. For $q>0$, their normalized versions occur as probabilities in the internal diffusion limited aggregation (IDLA) stochastic process. 4) A lift of Macdonald's reduced word identity in the $q$-Klyachko algebra. 5) The Schubert expansion of the Chow class of the standard split Deligne--Lusztig variety in type $A$, when $q$ is a prime power.

Motivation & Objective

• To extend classical Eulerian number identities to a q-deformed setting using algebraic and combinatorial structures.
• To provide a q-analog of the reduced word formula of Macdonald via a new generating function involving q-hook-lengths.
• To generalize the evaluation of certain symmetric functions to include a q-parameter, linking them to major index statistics on standard Young tableaux.
• To establish a q-deformed version of the interval property in the context of Schubert polynomials and quasisymmetric functions.

Proposed method

• Utilizes the alternative expression for $ a_w(q) $ from equation (LABEL:eq:awq_formula) and applies a preceding proposition to derive the main identity.
• Employs the sum over standard Young tableaux $ \mathrm{SYT}(\lambda, m-1) $ weighted by the major index $ \mathrm{maj}(T) $, scaled by a power of $ q $.
• Incorporates the q-hook-length $ h(i,j)_q = [\lambda_i + \lambda_j' - i - j + 1]_q $ for each cell $ (i,j) \in \lambda $.
• Applies the transformation $ q^{-N(\lambda)} $ to normalize the generating function, aligning it with known q-analog identities.
• Relies on results from [NT20, Corollary 6.18] and extends their framework to include the q-parameter in the context of m-Grassmannian permutations.
• Uses the interval property definition to characterize polynomials whose images under $ \pi $ or $ \pi^+ $ have only interval supports in the basis $ B $ or $ B^+ $.

Experimental results

Research questions

• RQ1 How can the classical reduced word formula of Macdonald be generalized to include a q-parameter?
• RQ2 What is the q-analog of the Eulerian number arising from the major index generating function over standard Young tableaux of a given shape?
• RQ3 Can the interval property in Schubert polynomials be extended to a q-deformed setting, and what does it imply for the structure of quasisymmetric functions?
• RQ4 How does the q-deformation of Klyachko's algebra relate to the hook-length formula and the major index statistic?

Key findings

• The generating function $ A_{(c_1,\dots,c_r)}(q) $ is expressed as $ q^{-N(\lambda)} \sum_{T \in \mathrm{SYT}(\lambda, m-1)} q^{\mathrm{maj}(T)} \prod_{(i,j) \in \lambda} h(i,j)_q $, providing a q-analog of Eulerian numbers.
• For the example with $ (c_1,\dots,c_5) = (0,1,2,1,1) $, the formula yields $ A_{01211}(q) = q^2(1+q)^2(1+q+q^2)^2(1+q^2) $, confirming the q-deformation.
• The q-hook-lengths $ h(i,j)_q $ are defined as $ [\lambda_i + \lambda_j' - i - j + 1]_q $, generalizing classical hook-lengths to the q-case.
• The result establishes a strong connection between m-Grassmannian permutations, standard Young tableaux, and q-specialized symmetric functions.
• The interval property is shown to hold for the images of certain polynomials under $ \pi $, suggesting deeper structural patterns in quasisymmetric and Schubert polynomials.