[Paper Review] A bialternant formula for odd symplectic characters and its application

Soichi Okada|Josai University Repository of Academia (Josai University)|May 30, 2019

Algebraic structures and combinatorial models6 references4 citations

TL;DR

This paper establishes a bialternant formula for odd symplectic characters—characters of indecomposable representations of odd symplectic groups—by introducing a novel $(n+1) \times (n+1)$ determinant formula involving variables $x_1,\dots,x_n$ and $z$. The formula generalizes Weyl's bialternant formula for classical groups and provides a linear algebraic proof of the Brent–Krattenthaler–Warnaar identity, an odd symplectic analogue of classical character identities.

ABSTRACT

We present a bialternant formula for odd symplectic characters, which are the characters of indecomposable modules of odd symplectic groups introduced by R. Proctor. As an application, we give a linear algebraic proof to an odd symplectic character identity due to R. P. Brent, C. Krattenthaler and S. O. Warnaar.

Motivation & Objective

• To develop a bialternant formula for odd symplectic characters, which are characters of indecomposable representations of non-reductive, odd-dimensional symplectic groups introduced by Proctor.
• To generalize Weyl's character formula, which applies to semisimple groups, to the case of odd symplectic groups that are neither semisimple nor reductive.
• To provide a linear algebraic proof of the Brent–Krattenthaler–Warnaar identity, a key character identity in odd symplectic representation theory.
• To bridge the gap between classical character formulas (e.g., Jacobi–Trudi, Cauchy) and their odd symplectic analogues by introducing a determinant-based character formula.

Proposed method

• Define the odd symplectic character $\mathrm{Sp}_{2n+1}(\lambda;x_1,\dots,x_n;z)$ via a generating function and a representation-free definition.
• Construct a matrix $A_\lambda$ of size $(n+1) \times (n+1)$ with entries involving $x_i^{\lambda_j + n - j + 2} - x_i^{-(\lambda_j + n - j + 2)}$ for $1 \leq i \leq n$, and $z^{\lambda_j + n - j + 1}$ for the last row.
• Derive the denominator $\det A_\varnothing$ as a product of terms corresponding to the root system between $C_n$ and $C_{n+1}$, involving $x_i - x_i^{-1}$ and $x_i^{1/2}x_j^{1/2} - x_i^{-1/2}x_j^{-1/2}$.
• Apply the Cauchy–Binet formula to the product of two such matrices to express the sum over partitions of a product of odd symplectic characters.
• Use determinant identities and factorizations involving Vandermonde-type determinants and $x_i^{2M+2}$-twisted terms to relate the sum to a single odd symplectic character of higher rank.
• Perform downward induction on $m$ to generalize the identity from $m = n$ to arbitrary $m \leq n$, using the substitution $x_1 = 0$ to reduce the rank.

Experimental results

Research questions

• RQ1 Can a bialternant formula be constructed for odd symplectic characters, analogous to Weyl’s formula for classical groups?
• RQ2 How does the character formula for odd symplectic groups differ from that of even symplectic groups, given the non-reductive nature of the odd symplectic group?
• RQ3 Is the Brent–Krattenthaler–Warnaar identity, which relates products of odd symplectic characters to a single higher-rank character, provable via linear algebraic methods rather than combinatorics?
• RQ4 What is the structure of the denominator in the bialternant formula for odd symplectic characters, and how does it relate to root systems of type $C_n$ and $C_{n+1}$?
• RQ5 Can the Cauchy-type identity for odd symplectic characters be generalized beyond the case $m = n$ using algebraic reduction techniques?

Key findings

• A bialternant formula for odd symplectic characters is established: $\mathrm{Sp}_{2n+1}(\lambda;x_1,\dots,x_n;z) = \frac{\det A_\lambda}{\det A_\varnothing}$, where $A_\lambda$ is a $(n+1) \times (n+1)$ matrix with mixed entries involving $x_i$-powers and $z$-powers.
• The denominator $\det A_\varnothing$ is explicitly computed as $\prod_{i=1}^n (x_i - x_i^{-1}) \prod_{1 \leq i < j \leq n+1} (x_i^{1/2}x_j^{1/2} - x_i^{-1/2}x_j^{-1/2})(x_i^{1/2}x_j^{-1/2} - x_i^{-1/2}x_j^{1/2})$, with $x_{n+1} = z$, matching the root system between $C_n$ and $C_{n+1}$.
• The Brent–Krattenthaler–Warnaar identity is proven using linear algebra: $\sum_{\lambda} z^{-r} \mathrm{Sp}_{2m+1}(\lambda;\mathbf{x};z) \mathrm{Sp}_{2n+1}((r^{n-m}) \cup \lambda;\mathbf{y};z) = \mathrm{Sp}_{2(m+n+1)}((r^{m+n+1});\mathbf{x},\mathbf{y},z)$.
• The proof relies on the Cauchy–Binet formula applied to matrices derived from the bialternant formula, with careful factor extraction and determinant manipulation.
• The identity is generalized from $m = n$ to arbitrary $m \leq n$ via downward induction, using the substitution $x_1 = 0$ to reduce the rank and preserve structure.
• The method provides a new, algebraic proof of the identity, bypassing the original combinatorial interpretation and establishing a direct link between determinant identities and odd symplectic character theory.