The Multiset Partition Algebra
Sridhar Narayanan, Digjoy Paul, and Shraddha Srivastava

TL;DR
This paper introduces the multiset partition algebra, establishes its Schur-Weyl duality with symmetric groups, and explores its structure, representations, and connections to other algebras, providing new tools for understanding symmetric functions and algebraic combinatorics.
Contribution
It generalizes the multiset partition algebra, proves its cellularity and semisimplicity under certain conditions, and develops an insertion algorithm for its module decomposition.
Findings
Establishes Schur-Weyl duality between $ ext{MP}_k(n)$ and $S_n$ actions.
Shows $ ext{MP}_ ext{lambda}( ext{ extbackslash xi})$ is a cellular algebra.
Provides generating functions for multiplicities of irreducible representations.
Abstract
We introduce the multiset partition algebra over , where is a field of characteristic and is a positive integer. When is specialized to a positive integer , we establish the Schur-Weyl duality between the actions of resulting algebra and the symmetric group on . The construction of generalizes to any vector of non-negative integers yielding the algebra over so that there is Schur-Weyl duality between the actions of and on . We find the generating function for the multiplicity of each irreducible representation of in , as varies, in terms of a plethysm of Schur functions. As consequences we obtain an indexing set for the…
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Taxonomy
TopicsCoding theory and cryptography · Advanced Algebra and Geometry · Advanced Combinatorial Mathematics
