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Enumeration of Anti-Invariant Subspaces and Touchard's Formula for the Entries of the $q$-Hermite Catalan Matrix

Amritanshu Prasad, Samrith Ram

Abstract

We express the number of anti-invariant subspaces for a linear operator on a finite vector space in terms of the number of its invariant subspaces. When the operator is diagonalizable with distinct eigenvalues, our formula gives a finite-field interpretation for the entries of the $q$-Hermite Catalan matrix. We also obtain an interesting new proof of Touchard's formula for these entries.

Enumeration of Anti-Invariant Subspaces and Touchard's Formula for the Entries of the $q$-Hermite Catalan Matrix

Abstract

We express the number of anti-invariant subspaces for a linear operator on a finite vector space in terms of the number of its invariant subspaces. When the operator is diagonalizable with distinct eigenvalues, our formula gives a finite-field interpretation for the entries of the -Hermite Catalan matrix. We also obtain an interesting new proof of Touchard's formula for these entries.
Paper Structure (6 sections, 13 theorems, 78 equations)

This paper contains 6 sections, 13 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Existence of a Universal Formula
  3. Determination of Coefficients in the Universal Formula
  4. Reduction to Heine's Transformations
  5. The $q$-Hermite Catalan matrix
  6. Acknowledgements

Key Result

Lemma 3

For each $T\in M_n(\mathbf F_q)$ and $0\leq a\leq n$,

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Lemma 3
  • proof
  • Proposition 4
  • proof
  • Corollary 5
  • proof
  • Theorem 6
  • Lemma 7
  • ...and 16 more