[Paper Review] Frobenius-Schur functions: summary of results

Grigori Olshanski, Amitai Regev+1 more|ArXiv.org|Mar 5, 2000

Algebraic structures and combinatorial models21 references3 citations

TL;DR

This paper introduces Frobenius–Schur functions, a new family of inhomogeneous symmetric functions that generalize Schur functions and provide an explicit determinantal formula for the dimension of skew Young diagrams in terms of Frobenius coordinates. The key contribution is a simple determinantal expression for these functions, along with combinatorial, generating series, Giambelli, and Sergeev–Pragacz-type formulas, all extending to a broader class of multiparameter Schur functions.

ABSTRACT

We introduce and study a family of inhomogeneous symmetric functions which we call the Frobenius-Schur functions. These functions are indexed by partitions and differ from the conventional Schur functions in lower terms only. Our interest in these new functions comes from the fact that they provide an explicit expression for the dimension of a skew Young diagram in terms of the Frobenius coordinates. This is important for the asymptotic theory of the characters of the symmetric groups. Our main result is a surprisingly simple determinantal expression of the Frobenius-Schur functions in terms of the conventional Schur functions. Other results include certain generating series, the Giambelli formula, vanishing properties and interpolation, a combinatorial formula (representation in terms of tableaux), and a Sergeev-Pragacz-type formula. Actually, we deal with a wider class of inhomogeneous symmetric functions which we call multiparameter Schur functions. These functions depend on an arbitrary doubly infinite sequence of parameters and interpolate between the Frobenius--Schur functions and the conventional Schur functions. This paper contains the statements of the results and the main formulas. Proofs will be given in an expanded version of the paper which will be posted in the arXiv.

Motivation & Objective

• To define and study Frobenius–Schur functions, a new class of inhomogeneous symmetric functions that generalize Schur functions.
• To provide an explicit expression for dim(ν/μ) in terms of the Frobenius coordinates of ν, addressing a problem from asymptotic character theory of symmetric groups.
• To establish a determinantal formula for these functions, extending known results in symmetric function theory.
• To generalize the framework to multiparameter Schur functions sμ;a, which reduce to Frobenius–Schur functions for specific parameter choices.
• To develop combinatorial, generating series, and interpolation formulas, and to prove a Sergeev–Pragacz-type formula for the new functions.

Proposed method

• The Frobenius–Schur functions FSμ are defined as inhomogeneous symmetric functions that differ from Schur functions sμ only in lower-degree terms.
• The functions are constructed via a parameterized family sμ;a, where a is a doubly infinite sequence; FSμ corresponds to a specific choice a_i = i - 1/2.
• A determinantal formula is derived using the structure of supersymmetric functions and the isomorphism between symmetric functions and supersymmetric functions via power sum specialization.
• A combinatorial formula is established using diagonal-strict tableaux, where each tableau entry contributes a factor fν;a(x,y) corresponding to the skew shape of the entry’s preimage.
• The Sergeev–Pragacz-type formula is proven via a Weyl-type character formula, involving antisymmetrization over S_n × S_n and a rational function gμ;a with products over x_i, y_j, and (x_i + y_j).
• The framework is shown to be compatible with the involution ω, and the results are extended to Schur’s Q-functions via work by Ivanov.

Experimental results

Research questions

• RQ1 How can the dimension of a skew Young diagram ν/μ be expressed explicitly in terms of the Frobenius coordinates of ν?
• RQ2 What is the structure of a new class of inhomogeneous symmetric functions that interpolate between Schur functions and the dimension of skew diagrams?
• RQ3 Can a determinantal formula be derived for these functions that generalizes known formulas in symmetric function theory?
• RQ4 How do the multiparameter Schur functions sμ;a behave under combinatorial, generating series, and interpolation constructions?
• RQ5 What is the analog of the Sergeev–Pragacz formula for this new class of functions, and how does it reduce to the classical case?

Key findings

• The Frobenius–Schur function FSμ provides a determinantal expression for dim(ν/μ) in terms of the Frobenius coordinates of ν, resolving a problem from asymptotic character theory of symmetric groups.
• A simple determinantal formula is established for FSμ in terms of conventional Schur functions, as stated in Theorem 9.
• A combinatorial formula is proven: sμ;a(x;y) equals the sum over all diagonal-strict tableaux of shape μ of products of fν;a(x_k,y_k) for each tableau level k.
• The Sergeev–Pragacz-type formula is derived for sμ;a(x;y) in terms of antisymmetrized rational functions involving (x_i|a)^{μ_i - i}, (y_i| frac{1}{2}a)^{μ'_i - i}, and products over (x_i + y_j).
• When a ≡ 0, the formula reduces to the classical Sergeev–Pragacz formula for Schur functions.
• The results extend to Schur’s Q-functions, with analogous formulas established by Vladimir Ivanov.