[Paper Review] Plethysm and the algebra of uniform block permutations

Rosa Orellana, Franco Saliola+2 more|arXiv (Cornell University)|Dec 27, 2021

Advanced Combinatorial Mathematics4 citations

TL;DR

This paper develops the representation theory of the uniform block permutation algebra (also known as the party algebra), a subalgebra of the partition algebra, by constructing irreducible representations via tableau models and establishing a Frobenius characteristic map to symmetric functions. The key contribution is linking the characters of this algebra to plethysms of Schur functions, specifically showing that the restriction of irreducible representations to the symmetric group yields characters given by the inner product of plethystic compositions $ s_{\lambda^{(1)}}[s_1] \cdots s_{\lambda^{(k)}}[s_k] $ with Schur functions.

ABSTRACT

We study the representation theory of the uniform block permutation algebra in the context of the representation theory of factorizable inverse monoids. The uniform block permutation algebra is a subalgebra of the partition algebra and is also known as the party algebra. We compute its characters and provide a Frobenius characteristic map to symmetric functions. This reveals connections of the characters of the uniform block permutation algebra and plethysms of Schur functions.

Motivation & Objective

• To develop the representation theory of the uniform block permutation algebra $\mathcal{U}_k$, a factorizable inverse monoid, using the general framework of finite inverse monoids.
• To characterize the irreducible representations of $\mathcal{U}_k$ via tableau models and Schützenberger representations.
• To define a Frobenius characteristic map from class functions on $\mathcal{U}_k$ to symmetric functions in multiple alphabets.
• To establish a precise connection between the characters of $\mathcal{U}_k$ and plethystic products of Schur functions.
• To compute the decomposition matrix of $\mathcal{U}_k$-representations restricted to symmetric groups, revealing coefficients tied to plethystic expansions.

Proposed method

• Uses the general theory of finite inverse monoids, including Clifford’s, Munn’s, and Ponizovskiĭ’s theorems, to analyze the structure of $\mathcal{U}_k$.
• Identifies maximal subgroups, $\mathscr{J}$-classes, and $\mathscr{L}$-classes to construct irreducible representations via Schützenberger representations.
• Introduces a Frobenius characteristic map $\phi_{\mathcal{U}_k}$ that maps class functions on $\mathcal{U}_k$ to elements in the $k$-fold tensor product of symmetric functions.
• Employs a generalized scalar product on symmetric functions over multiple alphabets $X_1, \dots, X_k$ to compute inner products of characters.
• Derives the character of an irreducible $\mathcal{U}_k$-representation as $\phi_{\mathcal{U}_k}(\chi^{\vec{\lambda}}_{\mathcal{U}_k}) = \mathbf{s}_{\vec{\lambda}}[\mathbf{E}]$, where $\mathbf{E}$ represents elementary symmetric functions in each alphabet.
• Uses the decomposition matrix $U_k$ to relate $\mathcal{U}_k$-representations to their restrictions to symmetric groups $\mathfrak{S}_k$, with entries given by inner products involving plethystic compositions.

Experimental results

Research questions

• RQ1 How can the irreducible representations of the uniform block permutation algebra $\mathcal{U}_k$ be constructed using combinatorial objects like set-valued tableaux?
• RQ2 What is the precise relationship between the characters of $\mathcal{U}_k$ and symmetric functions, particularly through a Frobenius characteristic map?
• RQ3 How do the irreducible representations of $\mathcal{U}_k$ decompose when restricted to the symmetric group $\mathfrak{S}_k$?
• RQ4 What is the structure of the decomposition matrix $U_k$ that relates $\mathcal{U}_k$-representations to $\mathfrak{S}_k$-representations, and what do its entries represent?
• RQ5 How do plethysms of Schur functions arise naturally in the character theory of $\mathcal{U}_k$, and what is their combinatorial interpretation?

Key findings

• The irreducible representations of $\mathcal{U}_k$ are indexed by $k$-tuples of partitions $\vec{\lambda} = (\lambda^{(1)}, \dots, \lambda^{(k)})$ satisfying $\sum_{i=1}^k i|\lambda^{(i)}| = k$, and admit a tableau model based on sequences of set-valued tableaux.
• The Frobenius characteristic map sends the character of an irreducible $\mathcal{U}_k$-representation to $\mathbf{s}_{\vec{\lambda}}[\mathbf{E}]$, a plethystic product of Schur functions $s_{\lambda^{(i)}}[s_i]$ in distinct alphabets.
• The restriction of an irreducible $\mathcal{U}_k$-representation to $\mathfrak{S}_k$ has character given by the inner product $\langle s_{\lambda^{(1)}}[s_1] \cdots s_{\lambda^{(k)}}[s_k], s_\mu \rangle$, which computes the multiplicity of the $\mathfrak{S}_k$-irreducible $V^\mu$ in the restriction.
• The decomposition matrix $U_k$ relating $\mathcal{U}_k$-representations to $\mathfrak{S}_k$-representations is upper unitriangular with entries corresponding to coefficients in the Schur expansion of plethystic products.
• For $k=2,3,4$, the matrices $U_k$ are explicitly computed, showing that entries can be greater than 1 (e.g., $U_{(\varnothing,(1),(1)),((3),(1))} = 2$), indicating nontrivial plethystic multiplicities.
• The matrix $U_k$ is upper unitriangular with $U_{\vec{\lambda},\vec{\lambda}} = 1$, and $U_{\vec{\lambda},\vec{\nu}} = 0$ if $\vec{\nu}$ has a smaller type than $\vec{\lambda}$ in the refined dominance order, confirming its triangular structure.