[Paper Review] Interpolation analogues of Schur Q-functions

Vladimir Ivanov|ArXiv.org|May 29, 2003

Advanced Combinatorial Mathematics24 references8 citations

TL;DR

This paper introduces multiparameter Schur Q-functions as interpolation analogues of classical Schur Q-functions, generalizing them via shifted factorial powers. It establishes key results including a combinatorial formula using marked shifted tableaux, a Nimmo-type Pfaffian identity, a Giambelli-Schur-type Pfaffian formula, generating functions for one- and two-row functions, and explicit transition coefficients between bases. The central contribution is a Pfaffian expression for the dimension of skew shifted Young diagrams.

ABSTRACT

We introduce interpolation analogues of Schur Q-functions - the multiparameter Schur Q-functions. We obtain for them several results: a combinatorial formula, generating functions for one-row and two-rows functions, vanishing and characterization properties, a Pieri-type formula, a Nimmo-type formula (a relation of two Pfaffians), a Giambelli-Schur-type Pfaffian formula, a determinantal formula for the transition coefficients between multiparameter Schur Q-functions with different parameters. We write an explicit Pfaffian expression for the dimension of skew shifted Young diagram. This paper is a continuation of author's paper math.CO/0303169 and is a partial projective analogue of the paper by A. Okounkov and G. Olshanski q-alg/9605042, and of the paper by G. Olshanski, A. Regev and A. Vershik math.CO/0110077.

Motivation & Objective

• To define and study multiparameter Schur Q-functions as interpolation analogues of classical Schur Q-functions.
• To establish combinatorial and algebraic formulas for these functions, generalizing known results for ordinary Schur Q-functions.
• To derive a Pfaffian expression for the dimension of skew shifted Young diagrams, linking to representation theory of spin-symmetric and Sergeev groups.
• To provide generating functions and transition coefficients between bases parameterized by different sequences $ (a_k) $ and $ (b_k) $.

Proposed method

• Define multiparameter Schur P- and Q-functions $ P_{\lambda;a} $ and $ Q_{\lambda;a} $ by replacing ordinary monomials with generalized powers $ (x - a_1)(x - a_2)\cdots(x - a_k) $, with $ a_1 = 0 $.
• Establish a Nimmo-type formula expressing $ P_{\lambda;a} $ as a ratio of two Pfaffians.
• Derive a combinatorial formula for $ Q_{\lambda;a} $ in terms of marked shifted tableaux, with an additional symmetry compared to the classical case.
• Use generating functions for one-row and two-row functions, proving new identities involving infinite products and generalized power series.
• Prove a Giambelli-Schur-type formula expressing $ Q_{\lambda;a} $ as a Pfaffian of two-row functions $ Q_{(\lambda_i,\lambda_j);a} $.
• Derive transition coefficients between bases $ \{Q_{\lambda;a}\} $ and $ \{Q_{\lambda;b}\} $ using symmetric function identities and the method of [ORV], leading to a determinant formula for the coefficients.

Experimental results

Research questions

• RQ1 How can Schur Q-functions be generalized to interpolation-type functions using multiparameter generalized powers?
• RQ2 What combinatorial and algebraic structures underlie the multiparameter Schur Q-functions, and how do they extend classical results?
• RQ3 Can a Pfaffian formula analogous to the Giambelli formula be established for these interpolation analogues?
• RQ4 What is the generating function for one-row and two-row multiparameter Schur Q-functions?
• RQ5 How do transition coefficients between different parameter sequences $ (a_k) $ and $ (b_k) $ behave, and can they be expressed explicitly?

Key findings

• A Nimmo-type formula expresses $ P_{\lambda;a} $ as a ratio of two Pfaffians, generalizing a classical identity.
• A combinatorial formula for $ Q_{\lambda;a} $ is given in terms of marked shifted tableaux, with an additional symmetry not present in the classical case.
• A generating function for one-row functions $ Q^{*}_{(r)} $ is derived: $ \sum_{r=0}^\infty \frac{Q^{*}_{(r)}}{(u\downarrow r)} = \prod_{i=1}^\infty \frac{u+1+x_i}{u+1-x_i} $.
• A generating function for two-row functions $ P_{(k,l);a} $ is established via Theorem 8.4, involving infinite products and a bilinear identity.
• A Giambelli-Schur-type Pfaffian formula expresses $ Q_{\lambda;a} $ as a Pfaffian of $ Q_{(\lambda_i,\lambda_j);a} $, generalizing Schur's original definition.
• An explicit Pfaffian expression for the dimension of a skew shifted Young diagram is given in Theorem 7.5, using factorial P-functions, and used to reprove Nazarov’s theorems on characters of the infinite spin-symmetric group.