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Projective geometries, $Q$-polynomial structures, and quantum groups

Paul Terwilliger

TL;DR

The paper extends the $Q$-polynomial framework for the projective geometry $L_N(q)$ by introducing a positive real parameter $\varphi$, recovering the known case at $\varphi=1$, and interpreting the resulting structure through the equitable presentation of $U_{q^{1/2}}(\mathfrak{sl}_2)$. It develops an explicit $A,A^*$ pair with eigenstructure $\theta_i=(\varphi q^{N-i}-q^i)/(q-1)$ and $\theta^*_i=q^{-i}$, and shows that $L_N(q)$ carries a $Q$-polynomial structure with respect to a distinguished vertex; it also embeds the standard module as a $U_{q^{1/2}}(\mathfrak{sl}_2)$-module and connects to Leonard systems of dual $q$-Krawtchouk type. The work further constructs four split decompositions—via variants $A^+,A^-,A_+,A_-$—that parallel the split decompositions in the theory of distance-regular graphs, enriching the interplay between quantum groups, Leonard pairs/systems, and projective geometries. These results provide a unified algebraic framework for analyzing spectral and combinatorial structures in $L_N(q)$ with potential implications for quantum algebra representations and combinatorial design theory.

Abstract

In 2023 we obtained a $Q$-polynomial structure for the projective geometry $L_N(q)$. In the present paper, we display a more general $Q$-polynomial structure for $L_N(q)$. Our new $Q$-polynomial structure is defined using a free parameter $\varphi$ that takes any positive real value. For $\varphi=1$ we recover the original $Q$-polynomial structure. We interpret the new $Q$-polynomial structure using the quantum group $U_{q^{1/2}}(\mathfrak{sl}_2)$ in the equitable presentation. We use the new $Q$-polynomial structure to obtain analogs of the four split decompositions that appear in the theory of $Q$-polynomial distance-regular graphs.

Projective geometries, $Q$-polynomial structures, and quantum groups

TL;DR

The paper extends the -polynomial framework for the projective geometry by introducing a positive real parameter , recovering the known case at , and interpreting the resulting structure through the equitable presentation of . It develops an explicit pair with eigenstructure and , and shows that carries a -polynomial structure with respect to a distinguished vertex; it also embeds the standard module as a -module and connects to Leonard systems of dual -Krawtchouk type. The work further constructs four split decompositions—via variants —that parallel the split decompositions in the theory of distance-regular graphs, enriching the interplay between quantum groups, Leonard pairs/systems, and projective geometries. These results provide a unified algebraic framework for analyzing spectral and combinatorial structures in with potential implications for quantum algebra representations and combinatorial design theory.

Abstract

In 2023 we obtained a -polynomial structure for the projective geometry . In the present paper, we display a more general -polynomial structure for . Our new -polynomial structure is defined using a free parameter that takes any positive real value. For we recover the original -polynomial structure. We interpret the new -polynomial structure using the quantum group in the equitable presentation. We use the new -polynomial structure to obtain analogs of the four split decompositions that appear in the theory of -polynomial distance-regular graphs.
Paper Structure (17 sections, 62 theorems, 114 equations)

This paper contains 17 sections, 62 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The projective geometry $L_N(q)$
  4. The matrices $\lbrace E^*_i \rbrace_{i=0}^N$ and the algebra $M^*$
  5. The matrix $A^*$
  6. The matrices $R, L$
  7. The matrix $A$ and the algebra $M$
  8. The algebra $T$
  9. The irreducible $T$-modules
  10. The eigenvalues of $A$
  11. The matrices $\lbrace E_i \rbrace_{i=0}^N$
  12. A $Q$-polynomial structure for $L_N(q)$
  13. $L_N(q)$ and the quantum group $U_{q^{1/2}} (\mathfrak{sl}_2)$
  14. $L_N(q)$ and Leonard systems
  15. The split decompositions of the standard module $V$
  16. ...and 2 more sections

Key Result

Lemma 3.2

(See axler.) For $x,y \in X$ we have

Theorems & Definitions (138)

  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Definition 4.3
  • ...and 128 more