There is a classical connection between the representation theory of the symmetric group and the general linear group called Schur-Weyl duality. Variations on this principle yield analogous connections between the symmetric group and other objects such as the partition algebra and more recently the multiset partition algebra. The partition algebra has a well-known basis indexed by graph-theoretic diagrams which allows the multiplication in the algebra to be understood visually as combinations of these diagrams. We construct an analogous basis for the multiset partition algebra called the diagram-like basis and use this basis to construct its irreducible representations and give a generating set. We also provide a change-of-basis formula from the orbit basis of the multiset partition algebra to this diagram-like basis which exhibits similarities to the analogous change-of-basis formula for the partition algebra.

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Keywords: representation theory, symmetric group, diagram algebra, partition algebra

Wilson, Alexander N. ^{1}

@article{ALCO_2024__7_4_1225_0, author = {Wilson, Alexander N.}, title = {A diagram-like basis for the multiset partition algebra}, journal = {Algebraic Combinatorics}, pages = {1225--1259}, publisher = {The Combinatorics Consortium}, volume = {7}, number = {4}, year = {2024}, doi = {10.5802/alco.364}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.364/} }

TY - JOUR AU - Wilson, Alexander N. TI - A diagram-like basis for the multiset partition algebra JO - Algebraic Combinatorics PY - 2024 SP - 1225 EP - 1259 VL - 7 IS - 4 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.364/ DO - 10.5802/alco.364 LA - en ID - ALCO_2024__7_4_1225_0 ER -

%0 Journal Article %A Wilson, Alexander N. %T A diagram-like basis for the multiset partition algebra %J Algebraic Combinatorics %D 2024 %P 1225-1259 %V 7 %N 4 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.364/ %R 10.5802/alco.364 %G en %F ALCO_2024__7_4_1225_0

Wilson, Alexander N. A diagram-like basis for the multiset partition algebra. Algebraic Combinatorics, Volume 7 (2024) no. 4, pp. 1225-1259. doi : 10.5802/alco.364. https://alco.centre-mersenne.org/articles/10.5802/alco.364/

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