Document Type
Honors Project On-Campus Access Only
Abstract
The irreducible modules of the symmetric group Sn are indexed by the integer partitions {λ : λ ⊢ n}. In the 1920's, Alfred Young defined representations on these modules according to the action of permutations σ in Sn on the standard Young tableaux of shape λ, denoted SYT(λ). In this paper, we solve an open problem by determining the change-of-basis matrix Aλ between two of these representations — the seminormal representation and the natural representation — by relating the entries of A λ to walks on the graph Γλ, which is the Hasse diagram for weak Bruhat order on SYT(λ). We then describe a recursive rule for computing these entries that puts the computational complexity for determining Aλ on the order of |SYT(λ)|2, and we abstract our formula to the Iwahori-Hecke algebra Hn(q) and the generalized symmetric group Gr,n.
Recommended Citation
Armon, Sam, "Transition Matrices for Young's Representations of the Symmetric Group" (2019). Mathematics, Statistics, and Computer Science Honors Projects. 44.
https://digitalcommons.macalester.edu/mathcs_honors/44
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Comments
A huge thanks to Professor Tom Halverson for advising me on this project.