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The lattice of submonoids of the uniform block permutations containing the symmetric group

Rosa Orellana, Franco Saliola, Anne Schilling, Mike Zabrocki

TL;DR

This work characterizes the intermediate monoids between the symmetric group $\mathfrak{S}_k$ and the uniform block permutation monoid $\mathcal{U}_k$ by linking submonoid structure to downsets in a new partial order $\preceq$ on partitions. It proves that the lattice of such submonoids is distributive under union and intersection, with a precise description: each submonoid corresponds to a downset of partition types, and explicit generators $e_{\pi}$ generate whole downward-closed families of $\mathscr{J}$-classes. The paper also connects representation theory to combinatorics by showing that $|J_\mu|$ equals a sum of squares of irreducible dimensions, and it notes that the poset is not Cohen–Macaulay, while providing enumerative consequences for the number of submonoids via antichains (OEIS A371505). Overall, the results give a clear, computable bridge between partition combinatorics and the algebraic structure of $\mathcal{U}_k$, with potential implications for restriction problems in related algebras and Hopf structures.

Abstract

We study the lattice of submonoids of the uniform block permutation monoid containing the symmetric group (which is its group of units). We prove that this lattice is distributive under union and intersection by relating the submonoids containing the symmetric group to downsets in a new partial order on integer partitions. Furthermore, we show that the sizes of the $\mathscr{J}$-classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra.

The lattice of submonoids of the uniform block permutations containing the symmetric group

TL;DR

This work characterizes the intermediate monoids between the symmetric group and the uniform block permutation monoid by linking submonoid structure to downsets in a new partial order on partitions. It proves that the lattice of such submonoids is distributive under union and intersection, with a precise description: each submonoid corresponds to a downset of partition types, and explicit generators generate whole downward-closed families of -classes. The paper also connects representation theory to combinatorics by showing that equals a sum of squares of irreducible dimensions, and it notes that the poset is not Cohen–Macaulay, while providing enumerative consequences for the number of submonoids via antichains (OEIS A371505). Overall, the results give a clear, computable bridge between partition combinatorics and the algebraic structure of , with potential implications for restriction problems in related algebras and Hopf structures.

Abstract

We study the lattice of submonoids of the uniform block permutation monoid containing the symmetric group (which is its group of units). We prove that this lattice is distributive under union and intersection by relating the submonoids containing the symmetric group to downsets in a new partial order on integer partitions. Furthermore, we show that the sizes of the -classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra.
Paper Structure (13 sections, 11 theorems, 41 equations, 2 figures)

This paper contains 13 sections, 11 theorems, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. A new partial order $\preceq$ on integer partitions
  4. Definition of a new partial order on integer partitions
  5. Comparison with refinement order
  6. Cover relations for $\preceq$
  7. Proof that the relations in Theorem \ref{['cover-relations']} are cover relations
  8. Proof that all cover relations for $\preceq$ are of the form specified in Theorem \ref{['cover-relations']}
  9. The submonoids of $\mathcal{U}_k$ containing $\mathfrak{S}_k$
  10. Concluding remarks
  11. The sizes of $\mathscr{J}$-classes are sums of squares of dimensions of irreducibles
  12. The order $\preceq$ on partitions is not Cohen--Macaulay
  13. The number of submonoids of $\mathcal{U}_k$ that contain $\mathfrak{S}_k$

Key Result

Theorem 1.1

For $k$ a positive integer, the set of submonoids of $\mathcal{U}_k$ that contain $\mathfrak{S}_k$ forms a distributive lattice with the operations of union and intersection. (Corollary cor:submonoid characterization) Define a partial order on partitions $\mu, \lambda$ of $k$ that are not equal to $ for some down set $I$ of the partial order $\preceq$.

Figures (2)

  • Figure 1: Hasse diagram for $(\mathcal{P}_6 \setminus \{1^6\}, \preceq)$.
  • Figure 2: The distributive lattices of submonoids of $\mathcal{U}_k$ containing the symmetric group $\mathfrak{S}_k$ for $k=4,5,6$. Each submonoid is labeled by a box containing an antichain of the poset $(\mathcal{P}_k \backslash \{1^k\},\preceq)$. The antichains of $\preceq$ are in bijection with the downsets of this poset (see Stanley.ECI) and in turn are in bijection with the submonoids by Corollary \ref{['cor:submonoid characterization']}. Below each box is the size of the corresponding submonoid.

Theorems & Definitions (27)

  • Theorem 1.1: Theorem \ref{['union-is-submonoid']}
  • Example 2.1
  • Example 2.2
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • Lemma 3.5
  • proof
  • ...and 17 more