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Durfee rectangle identities as character identities for infinite Fibonacci configurations

Timur Kenzhaev

Abstract

We introduce a natural generalization of Maya diagrams -- the space of infinite Fibonacci configurations, which are specified functions on $\mathbb{Z}$ with values $1$ and $0$. Infinite Fibonacci configurations are particularly interesting as soon as they parametrize Feigin-Stoyanovsky type bases in lattice vertex superalgebras $V_{\sqrt{N}\mathbb{Z}}$ and their irreducible modules. We calculate the character of such configurations space by two different ways and obtain series of combinatorial identities. These identities turn out to be Durfee rectangle identities with shifts in base and height.

Durfee rectangle identities as character identities for infinite Fibonacci configurations

Abstract

We introduce a natural generalization of Maya diagrams -- the space of infinite Fibonacci configurations, which are specified functions on with values and . Infinite Fibonacci configurations are particularly interesting as soon as they parametrize Feigin-Stoyanovsky type bases in lattice vertex superalgebras and their irreducible modules. We calculate the character of such configurations space by two different ways and obtain series of combinatorial identities. These identities turn out to be Durfee rectangle identities with shifts in base and height.
Paper Structure (11 sections, 2 theorems, 91 equations, 14 figures)

This paper contains 11 sections, 2 theorems, 91 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Fibonacci configurations
  3. Fibonacci-$1$ configurations
  4. Fibonacci-$l$ configurations
  5. Infinite Fibonacci configurations
  6. Character identities
  7. $l = 0$ case
  8. $l = 1$ case
  9. General case
  10. Lattice vertex algebras
  11. Images

Key Result

Theorem 1.1

$L_{(0 , 1)}$ has the basis of semi-infinite monomials $e_{i_1}\,e_{i_2}\,e_{i_3}\dots$ such that $L_{(1, 1)}$ has the basis of semi-infinite monomials $e_{i_1}\,e_{i_2}\,e_{i_3}\dots$ such that

Figures (14)

  • Figure 1: Illustration of \ref{['qBinom1']}: any Young diagram corresponding to the partition into m distinct $\leq N$ parts can be obtained by appending Young diagram contained in the $m\times (N - m)$ rectangle to the $(1, 2, \ldots, m)$ shape from the right.
  • Figure 2: Illustration of \ref{['qBinom2']}: any Young diagram corresponding to the partition into $m$ distinct $\leq~N~-~1$ parts with adjacent differing $\geq 2$ can be obtained by appending Young diagram contained in the $m\times (N - 2m)$ rectangle to the $(1, 3, \ldots, 2m - 1)$ shape from the right.
  • Figure 3: Identity \ref{['DurfeeRectangleIdentity0']} at $s = 0$. Durfee rectangles $k\times k$
  • Figure 4: Identity \ref{['DurfeeRectangleIdentity0']} at $s = 1$. Durfee rectangles $k\times (k + 1)$
  • Figure 5: Identity \ref{['DurfeeRectangleIdentity0']} at $s = 2$. Durfee rectangles $k\times (k + 2)$
  • ...and 9 more figures

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 3.1
  • Remark 3.2
  • Conjecture 4.1