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Hook-Lengths, Symplectic/Orthogonal Contents and Amdeberhan's Conjectures

Chenglang Yang

TL;DR

This work proves the general Amdeberhan conjectures linking symplectic/orthogonal contents of partitions to infinite-product generating functions, thereby extending Nekrasov--Okounkov and Stanley hook-content-type formulas to symplectic and orthogonal contexts. The authors deploy vertex operators to construct and analyze partition functions built from (double) symplectic Schur functions, deriving explicit product formulas and Cauchy-type identities. Key results include explicit generating-function formulas for sums of $q^{|oldsymbollank|}$ weighted by $(t+c_{sp}(u))/h(u)$ and $(t+c_O(u))/h(u)$, their dualities, and extensions to mixed content settings. These findings deepen the representation-theoretic interpretation of partition statistics and provide broadly applicable tools for related combinatorial and mathematical-physics problems.

Abstract

The symplectic/orthogonal contents of partitions are related to the dimensions of irreducible representations of symplectic/orthogonal groups. In 2012, motivated by Nekrasov--Okounkov's hook-length formula and Stanley's hook-content formula, Amdeberhan proposed several conjectures about infinite product formulas for certain generating functions of hook-lengths and symplectic/orthogonal contents. Some special cases of his conjectures were recently proved by Amdeberhan, Andrews and Ballantine. In this paper, we prove the general cases of Amdeberhan's conjectures.

Hook-Lengths, Symplectic/Orthogonal Contents and Amdeberhan's Conjectures

TL;DR

This work proves the general Amdeberhan conjectures linking symplectic/orthogonal contents of partitions to infinite-product generating functions, thereby extending Nekrasov--Okounkov and Stanley hook-content-type formulas to symplectic and orthogonal contexts. The authors deploy vertex operators to construct and analyze partition functions built from (double) symplectic Schur functions, deriving explicit product formulas and Cauchy-type identities. Key results include explicit generating-function formulas for sums of weighted by and , their dualities, and extensions to mixed content settings. These findings deepen the representation-theoretic interpretation of partition statistics and provide broadly applicable tools for related combinatorial and mathematical-physics problems.

Abstract

The symplectic/orthogonal contents of partitions are related to the dimensions of irreducible representations of symplectic/orthogonal groups. In 2012, motivated by Nekrasov--Okounkov's hook-length formula and Stanley's hook-content formula, Amdeberhan proposed several conjectures about infinite product formulas for certain generating functions of hook-lengths and symplectic/orthogonal contents. Some special cases of his conjectures were recently proved by Amdeberhan, Andrews and Ballantine. In this paper, we prove the general cases of Amdeberhan's conjectures.
Paper Structure (13 sections, 14 theorems, 92 equations)

This paper contains 13 sections, 14 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schur functions and symplectic Schur functions
  4. Inner product and the vacuum expectation value
  5. Vertex operator
  6. A partition function consisting of symplectic Schur functions
  7. A partition function of symplectic Schur functions
  8. Proof of Theorem \ref{['thm:sum sp as p_k']}
  9. Hook-symplectic/orthogonal contents formulas
  10. A Cauchy-identity type formula for symplectic Schur functions
  11. Partition function of double symplectic Schur functions
  12. Proofs of Theorems \ref{['thm:main csp^2 cO^2']} \ref{['thm:main csp*c']} and Corollary \ref{['cor:main csp^2=csp']}
  13. Conflict of interest and data availability statement

Key Result

Theorem 1.1

For indeterminates $t, q$, we have where $\mathcal{P}$ is the set of partitions.

Theorems & Definitions (17)

  • Theorem 1.1: =Conjecture 6.2 (a) in Amd12
  • Corollary 1.2: =Conjecture 6.2 (b) in Amd12
  • Theorem 1.3
  • Corollary 1.4: =Conjecture 6.3 (c) in Amd12
  • Theorem 1.5
  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 7 more