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Party-Hecke algebras

Diego Arcis, Jesús Juyumaya

Abstract

Party-Hecke algebras are introduced as a two-parameter deformation of party algebras, where one parameter deforms the party generators and the other deforms the elementary transpositions. We construct a basis for this algebra and show that it can be realized as a quotient of the algebra of braids and ties. Furthermore, we study the party monoid and its relationship with the tied symmetric monoid and their associated algebras.

Party-Hecke algebras

Abstract

Party-Hecke algebras are introduced as a two-parameter deformation of party algebras, where one parameter deforms the party generators and the other deforms the elementary transpositions. We construct a basis for this algebra and show that it can be realized as a quotient of the algebra of braids and ties. Furthermore, we study the party monoid and its relationship with the tied symmetric monoid and their associated algebras.
Paper Structure (28 sections, 26 theorems, 95 equations, 14 figures)

This paper contains 28 sections, 26 theorems, 95 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Set partitions
  4. The partition monoid
  5. The symmetric group and the braid group
  6. Ramified monoids
  7. Twisted monoid algebras and the partition algebra
  8. The Iwahori--Hecke algebra and the algebra of braids and ties
  9. Party monoids and algebras
  10. The party monoid and the party algebra
  11. Maximal subgroups and cellular structure
  12. Maximal subgroups of the party monoid
  13. Maximal subgroups of the tied symmetric monoid
  14. Cellularity of the party algebra
  15. A twisted monoid algebra of the tied braid monoid
  16. ...and 13 more sections

Key Result

Proposition 2.1

Let $I=(I_1,\ldots,I_k)$ be a set partition of $\ldbrack 1,n\rdbrack$. Then $I=f_{I_1}\cdots f_{I_k}$. More precisely, for $B=\{i_1<\cdots<i_k\}\subseteq\ldbrack 1,n\rdbrack$ with $k=|B|>1$, we have:

Figures (14)

  • Figure 1: Strand diagram of a set partition.
  • Figure 2: The 5 elements of $P_3$.
  • Figure 3: The 6 elements of $\mathfrak{S}_3$.
  • Figure 4: Generator $(1,f_{i,j})=e_{i,j}.$
  • Figure 5: Generator $t_i$.
  • ...and 9 more figures

Theorems & Definitions (59)

  • Proposition 2.1: ArJu2021
  • Remark 2.2
  • Theorem 2.3: East2011
  • Proposition 3.1
  • proof
  • Proposition 3.2: Normal form
  • proof
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • ...and 49 more