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Vanishing Immanants

Hassan Cheraghpour, Bojan Kuzma

Abstract

We classify all the irreducible characters of a symmetric group such that the induced immanant function $d_χ$ vanishes identically on alternate matrices with the entries in the complex field.

Vanishing Immanants

Abstract

We classify all the irreducible characters of a symmetric group such that the induced immanant function vanishes identically on alternate matrices with the entries in the complex field.
Paper Structure (7 sections, 22 theorems, 52 equations, 3 figures)

This paper contains 7 sections, 22 theorems, 52 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Destructibility of diagrams
  4. Two rules which preserve the (in)destructibility of diagrams
  5. Characters induced by (in)destructible diagrams
  6. Addendum
  7. Figures

Key Result

Proposition 1.1

Let $A \in \mathbb{A}_{n}({\mathbb C})$. Then $d_{\chi}(A) =\sum_{\sigma \in P_{n}}\chi(\sigma) \prod_{i=1}^{n} a_{i\sigma(i)}$.

Figures (3)

  • Figure 1: An example of an indestructible diagram (see Lemma \ref{['Triangle-Remove']}).
  • Figure 2: All six different tilings of diagram $(4,4,3,1)$. Notice that the number of vertical dominoes is always odd (c.f. Lemma \ref{['parity2']}). Notice also that each tiling contains at least one (shaded) domino rim-hook (c.f. Lemma \ref{['pattern']}).
  • Figure 3: There are $|T|=7$ possibilities to recursively remove rim-hook dominoes by following a fixed tiling $T$ of diagram $(4,4,3,1)$ (c.f. Theorem \ref{['character']} and the text immediately before it). The domino rim-hooks of each obtained diagram are shaded.

Theorems & Definitions (42)

  • Proposition 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • Lemma 3.5
  • proof
  • ...and 32 more