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Iterative construction of $\mathfrak{S}_p \times \mathfrak{S}_p$ group-adapted irreducible matrix units for the walled Brauer algebra

Michał Horodecki, Michał Studziński, Marek Mozrzymas

Abstract

In this work, we present an algorithmic treatment of the representation theory of the algebra of partially transposed permutation operators, denoted by $\mathcal{A}^d_{p,p}$, which is a matrix representation of the abstract walled Brauer algebra. We provide an explicit and fully developed framework for constructing irreducible matrix units within the algebra. In contrast to the established earlier Gelfand-Tsetlin type constructions, the presented matrix units are adapted to the action of the subalgebra $\mathbb{C}[\mathfrak{S}_p] \times \mathbb{C}[\mathfrak{S}_p]$, where $\mathfrak{S}_p$ is the symmetric group. What is more, the basis is constructed in such a way that it produces the decomposition of the algebra into a direct sum of ideals, in contrast to its nested structure considered before. The decomposition of this kind has not been considered before in full generality. Our method reveals a recursive scheme for generating irreducible matrix units in all ideals of $\mathcal{A}^d_{p,p}$, offering a systematic approach that applies to small system sizes and arbitrary local dimensions. We apply the developed formalism to the algebra $\mathcal{A}^d_{2,2}$ and illustrate the algorithm in practice. In addition, using the constructed basis, we proved a novel contraction theorem for the elements from $\mathcal{A}^d_{3,3}$, which is the starting point for further investigations.

Iterative construction of $\mathfrak{S}_p \times \mathfrak{S}_p$ group-adapted irreducible matrix units for the walled Brauer algebra

Abstract

In this work, we present an algorithmic treatment of the representation theory of the algebra of partially transposed permutation operators, denoted by , which is a matrix representation of the abstract walled Brauer algebra. We provide an explicit and fully developed framework for constructing irreducible matrix units within the algebra. In contrast to the established earlier Gelfand-Tsetlin type constructions, the presented matrix units are adapted to the action of the subalgebra , where is the symmetric group. What is more, the basis is constructed in such a way that it produces the decomposition of the algebra into a direct sum of ideals, in contrast to its nested structure considered before. The decomposition of this kind has not been considered before in full generality. Our method reveals a recursive scheme for generating irreducible matrix units in all ideals of , offering a systematic approach that applies to small system sizes and arbitrary local dimensions. We apply the developed formalism to the algebra and illustrate the algorithm in practice. In addition, using the constructed basis, we proved a novel contraction theorem for the elements from , which is the starting point for further investigations.

Paper Structure

This paper contains 12 sections, 21 theorems, 82 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Dedication
  2. Introduction
  3. Basic facts and notations used throughout the manuscript
  4. Comparison with the known literature and our results
  5. The algorithm
  6. Decomposition of the algebra $\mathcal{A}^d_{p,p}$ into direct sum of $\widetilde{\mathcal{M}}^{(p)}$ and $\widetilde{\mathcal{M}}^{(p-1)}$
  7. Examples of irreducible matrix units construction for $\mathcal{A}^d_{2,2}$
  8. Further properties of the algebra $\mathcal{A}^d_{p,p}$ and irreducible matrix units construction for $p=3$.
  9. Additional general considerations concerning the algebra $\mathcal{A}^d_{p,p}$
  10. Construction of irreducible matrix units for $\mathcal{A}^d_{3,3}$
  11. Discussion
  12. Appendix

Key Result

Lemma 1

For an arbitrary operator $X\in \mathcal{A}_{p,p}^{d}$ and operator $V^{(r)}$, where $1\leq r\leq p$, given through equation eq:Vl the following equality holds: where $X_{p\setminus r, p\setminus r}\in \mathcal{A}^{d}_{p\setminus r, p\setminus r}$, and the symbol $p\setminus r$ means that the operators act non-trivially everywhere except the first $r$ arcs, counting from the wall.

Figures (3)

  • Figure 1: The walled Brauer algebra$\mathcal{B}^{\delta}_{m,n}$, where $m,n \geq 0$, and $\delta \in \mathbb{C}$, was introduced in a series of works including Bra37VGTuraev_1990KOIKE198957BENKART1994529BEN96bulgakova:tel-02554375Cox1. This algebra is defined abstractly as a linear span over $\mathbb{C}$ of certain diagrammatic elements. Each diagram consists of two rows of $m + n$ nodes, separated by a vertical wall placed between the first $m$ and the remaining $n$ nodes. Edges connect nodes in pairs, subject to the following rules: (a) Pairs of nodes within the same row must lie on opposite sides of the wall. (b) Pairs of nodes between different rows must lie on the same side of the wall. This combinatorial structure defines the multiplication in the algebra via concatenation of diagrams, with closed loops contributing a scalar factor of $\delta$. The dimension of $\mathcal{B}^{\delta}_{m,n}$ coincides with the number of such valid pairings, and in particular equals $(m+n)!$, matching the size of the symmetric group $\mathfrak{S}_{m+n}$. Here we present an example of graphical composition of two diagrams $b_1,b_2 \in \mathcal{B}_{4,4}^\delta$. Identifying a closed loop (in red) results in multiplying the diagram by a scalar $\delta \in \mathbb{C}$. We see that the composition $b_1 \circ b_2$ remains within $\mathcal{B}_{4,4}^\delta$.
  • Figure 2: Diagram representation of the operator $V^{(r)}$ from equation \ref{['eq:Vl']}. On the abstract level, this object is an element of the walled Brauer algebra $\mathcal{B}_{p,p}^\delta$ with $\delta=d$. On the representation space $(\mathbb{C}^d)^{\otimes 2p}$, the operator $V^{(r)}$ is an element of the algebra of the partially transposed permutation operators $\mathcal{A}_{p,p}^{d}$.
  • Figure 3: Graphic presents relation between non-trivial action of $V^{(r)},V^{(s)}$, and $V^{(s\setminus r)}$. The red arcs represent the domain for the operator $V^{(r)}$. The red arcs, together with the blue arcs, give us a domain for $V^{(s)}$. From this, it is easy to deduce that the blue arcs are for $V^{(s\setminus r)}$. For the clarity of the figure, we draw here only the bottom part of the considered elements.

Theorems & Definitions (26)

  • Lemma 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • Corollary 5
  • Proposition 6
  • Corollary 7
  • Proposition 8
  • Definition 9
  • Proposition 10
  • ...and 16 more