[Paper Review] A q-analog of Schur's Q-functions

Geanina Tudose, Mike Zabrocki|ArXiv.org|Mar 5, 2002

Algebraic structures and combinatorial models20 references3 citations

TL;DR

This paper introduces a $q$-analog of Schur's $Q$-functions, denoted $G_{\lambda}[X;q]$, by applying a $q$-twisting operation to the operator that generates $Q$-functions, analogous to how Hall-Littlewood symmetric functions arise from Schur functions. The key result is that the change-of-basis coefficients between $G_{\mu}[X;q]$ and $Q_{\lambda}[X]$ are polynomials in $q$ with non-negative integer coefficients, which at $q=1$ count marked shifted tableaux, generalizing the role of Kostka-Foulkes polynomials in the symmetric function setting.

ABSTRACT

We present a family of analogs of the Hall-Littlewood symmetric functions in the $Q$-function algebra. The change of basis coefficients between this family and Schur's $Q$-functions are $q$-analogs of numbers of marked shifted tableaux. These coefficients exhibit many parallel properties to the Kostka-Foulkes polynomials.

Motivation & Objective

• To define a $q$-analog of Hall-Littlewood symmetric functions within the $Q$-function algebra $\Gamma$, which is generated by odd power sums.
• To establish that the change-of-basis coefficients between the new family $G_{\lambda}[X;q]$ and Schur's $Q$-functions are polynomials in $q$ with non-negative integer coefficients.
• To demonstrate that these coefficients generalize the Kostka-Foulkes polynomials by reducing to the number of marked shifted tableaux at $q=1$.
• To explore whether combinatorial tools like the RSK algorithm or jeu de taquin can be extended to interpret these $q$-analog coefficients.
• To identify structural parallels between the new $G_{\lambda}[X;q]$ functions and Hall-Littlewood functions, suggesting a deeper algebraic or representation-theoretic principle.

Proposed method

• Define the $q$-twisted operator $\widetilde{{\bf Q}_m}^q$ from the operator ${\bf Q}_m$ that generates $Q$-functions via degree-raising recursion.
• Construct $G_{\lambda}[X;q]$ recursively using $G_{(m,\lambda_1,\dots)}[X;q] = \widetilde{{\bf Q}_m}^q(G_{\lambda}[X;q])$ for $m > \lambda_1$.
• Use the operator framework to derive recurrence relations analogous to those of Hall-Littlewood functions in the symmetric function algebra.
• Analyze the transition matrix between $G_{\mu}[X;q]$ and $Q_{\lambda}[X]$ to show that coefficients are $q$-analogs of marked shifted tableau counts.
• Provide explicit tables of coefficients $2^{\ell(\lambda)-\ell(\mu)}L_{\lambda\mu}(q)$ for $n=4$ to $9$, illustrating the polynomial nature and non-negative coefficients.
• Conjecture that the coefficient statistics are independent of labeling choices (e.g., $k^*$ vs. $k$) in tableau posets, based on rank function patterns.

Experimental results

Research questions

• RQ1 What is the correct $q$-analog of Hall-Littlewood symmetric functions within the $Q$-function algebra $\Gamma$?
• RQ2 Do the change-of-basis coefficients between $G_{\mu}[X;q]$ and $Q_{\lambda}[X]$ generalize the Kostka-Foulkes polynomials in a combinatorial and algebraic sense?
• RQ3 Can the $q$-twisting operation $\widetilde{{\bf Q}_m}^q$ be given a combinatorial, geometric, or representation-theoretic interpretation?
• RQ4 Is there a combinatorial model—such as a generalized RSK or jeu de taquin—for interpreting the coefficients of $G_{\mu}[X;q]$ in terms of marked shifted tableaux?
• RQ5 How do the properties of $G_{\lambda}[X;q]$ compare to those of Hall-Littlewood functions, and what does this suggest about the universality of $q$-twisting?

Key findings

• The $q$-analog $G_{\lambda}[X;q]$ is defined via a $q$-twisted operator $\widetilde{{\bf Q}_m}^q$, extending the recursive generation of $Q$-functions.
• The coefficients in the expansion $G_{\mu}[X;q] = \sum_{\lambda} L_{\lambda\mu}(q) Q_{\lambda}[X]$ are polynomials in $q$ with non-negative integer coefficients.
• At $q=1$, the coefficients $L_{\lambda\mu}(1)$ count the number of marked shifted tableaux of shape $\lambda$ and content $\mu$.
• Explicit tables for $n=4$ to $9$ confirm that $2^{\ell(\lambda)-\ell(\mu)}L_{\lambda\mu}(q)$ is a polynomial with non-negative coefficients, supporting the conjecture of positivity.
• The structure of the coefficient matrices resembles that of Hall-Littlewood functions, with upper-triangular form and increasing powers of $q$ along diagonals.
• A conjectured poset of marked shifted tableaux for shape $(4,3,2)$ suggests that the coefficient statistics may be independent of labeling choices (e.g., $k^*$ vs. $k$), though the covering relation remains unknown.