[Paper Review] Doubly Symmetric Functions

Allan Berele, Bridget Eileen Tenner|ArXiv.org|Mar 30, 2009

Advanced Combinatorial Mathematics7 references3 citations

TL;DR

This paper introduces doubly symmetric functions—symmetric functions that coincide under evaluation as Schur functions and hook Schur functions for any split of variables. The key result is that this subalgebra of symmetric functions is generated by odd power sum symmetric functions, and a Schur function is doubly symmetric if and only if its partition is a staircase shape.

ABSTRACT

In this paper we introduce doubly symmetric functions, arising from the equivalence of particular linear combinations of Schur functions and hook Schur functions. We study algebraic and combinatorial aspects of doubly symmetric functions, in particular as they form a subalgebra of the algebra of symmetric functions. This subalgebra is generated by the odd power sum symmetric functions. One consequence is that a Schur function itself is doubly symmetric if and only if it is the Schur function of a staircase shape.

Motivation & Objective

• To define and study a new class of symmetric functions, called doubly symmetric functions, arising from the coincidence of Schur and hook Schur functions under variable partitioning.
• To characterize the algebraic structure of the subalgebra formed by these functions within the ring of symmetric functions.
• To determine the basis and dimension of the space of doubly symmetric functions of a given degree.
• To establish a connection between doubly symmetric functions and partitions into odd or distinct parts.
• To prove that a Schur function is doubly symmetric if and only if its indexing partition is a staircase shape.

Proposed method

• Define doubly symmetric functions via the equality of Schur and hook Schur function evaluations across any split of variables into two sets.
• Introduce the doubly Schur function $ DS_{ ho} = D(S_{ ho}) $, where $ D $ is a surjective homomorphism from the ring of symmetric functions to the subalgebra $ olinebreakigskip\mathcal{D} $ of doubly symmetric functions.
• Show that $ D $ acts as an eigenvector map on power sum symmetric functions: $ D(p_{ ho}) = 0 $ if $ ho $ has an even part, and $ D(p_{ ho}) = 2^{\text{ht}(\rho)}p_{\rho} $ if all parts of $ \rho $ are odd.
• Prove that the kernel of $ D $ is the ideal $ \mathcal{I} $ generated by $ S_{\lambda} - S_{\lambda'} $, leading to $ \mathcal{D} \simeq \mathcal{S}/\mathcal{I} $.
• Establish that $ \mathcal{D} $ is the orthogonal complement of $ \mathcal{I} $ under the standard inner product, so $ \mathcal{S} = \mathcal{D} \oplus \mathcal{I} $.
• Use generating functions to show that the dimension of the space of doubly symmetric functions of degree $ n $ equals the number of partitions of $ n $ into odd parts, or equivalently into distinct parts.

Experimental results

Research questions

• RQ1 What conditions ensure that a symmetric function is doubly symmetric, i.e., equal to its hook Schur function evaluation under any variable split?
• RQ2 What is the algebraic structure of the subalgebra $ \mathcal{D} $ of doubly symmetric functions within the ring of symmetric functions?
• RQ3 Which Schur functions are doubly symmetric, and what is the combinatorial characterization of their indexing partitions?
• RQ4 How are the dimensions of the spaces of doubly symmetric functions of degree $ n $ related to partition functions into odd or distinct parts?
• RQ5 What is the image and kernel of the homomorphism $ D $ mapping Schur functions to doubly Schur functions?

Key findings

• The subalgebra $ \mathcal{D} $ of doubly symmetric functions is generated by the odd power sum symmetric functions $ p_n $ with $ n $ odd.
• The dimension of the space of doubly symmetric functions of degree $ n $ is equal to the number of integer partitions of $ n $ into odd parts, given by the coefficient of $ x^n $ in $ \prod_{k \geq 0} (1 - x^{2k+1})^{-1} $.
• Equivalently, this dimension equals the number of partitions of $ n $ into distinct parts, matching the coefficient of $ x^n $ in $ \prod_{k \geq 1} (1 + x^k) $.
• A Schur function $ S_{\lambda} $ is doubly symmetric if and only if $ \lambda $ is a staircase shape, i.e., $ \lambda = (m, m-1, \dots, 1) $ for some $ m $.
• The doubly Schur functions $ \{ DS_{\lambda} \} $ span $ \mathcal{D} $, though they are not linearly independent, and $ D $ is surjective with kernel $ \mathcal{I} $.
• The map $ D $ sends $ p_{\lambda} $ to 0 if $ \lambda $ has any even part, and to $ 2^{\text{ht}(\lambda)} p_{\lambda} $ if all parts of $ \lambda $ are odd.