Tridiagonal pairs, alternating elements, and distance-regular graphs

DOI10.1016/J.JCTA.2022.105724arXiv2207.07741MaRDI QIDQ6405130

Publication date: 15 July 2022

Abstract: The positive part

U_{q}^{+}

of

U_{q} (h a t {m a t h f r a k s l}_{2})

has a presentation with two generators

W_{0}

,

W_{1}

and two relations called the

q

-Serre relations. The algebra

U_{q}^{+}

contains some elements, said to be alternating. There are four kinds of alternating elements, denoted

l b r a c e W_{- k} b r a c e_{k i n m a t h b b N}

,

l b r a c e W_{k + 1} b r a c e_{k i n m a t h b b N}

,

l b r a c e G_{k + 1} b r a c e_{k i n m a t h b b N}

,

l b r a c e {i l d e G}_{k + 1} b r a c e_{k i n m a t h b b N}

. The alternating elements of each kind mutually commute. A tridiagonal pair is an ordered pair of diagonalizable linear maps

A, A^{*}

on a nonzero, finite-dimensional vector space

V

, that each act in a (block) tridiagonal fashion on the eigenspaces of the other one. Let

A

,

A^{*}

denote a tridiagonal pair on

V

. Associated with this pair are six well-known direct sum decompositions of

V

; these are the eigenspace decompositions of

A

and

A^{*}

, along with four decompositions of

V

that are often called split. In our main results, we assume that

A

,

A^{*}

has

q

-Serre type. Under this assumption

A

,

A^{*}

satisfy the

q

-Serre relations, and

V

becomes an irreducible

U_{q}^{+}

-module on which

W_{0} = A

and

W_{1} = A^{*}

. We describe how the alternating elements of

U_{q}^{+}

act on the above six decompositions of

V

. We show that for each decomposition, every alternating element acts in either a (block) diagonal, (block) upper bidiagonal, (block) lower bidiagonal, or (block) tridiagonal fashion. We investigate two special cases in detail. In the first case the eigenspaces of

A

and

A^{*}

all have dimension one. In the second case

A

and

A^{*}

are obtained by adjusting the adjacency matrix and a dual adjacency matrix of a distance-regular graph that has classical parameters and is formally self-dual.

Mathematics Subject Classification ID

Combinatorial aspects of representation theory (05E10) Quantum groups (quantized enveloping algebras) and related deformations (17B37) Association schemes, strongly regular graphs (05E30)

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