Honors Project On-Campus Access Only
The object of our study is a bijective algorithm that turns a word in the k-dimensional positive integer lattice into a sequence of standard Young tableaux, where the subsequence of every other tableaux, beginning with the first entry in said sequence, is a sequence of standard Young tableaux on n boxes, and each successive tableau is gotten by deleting a box and reinserting it. The tableaux in these subsequences index bases of irreducible modules for the symmetric group of degree n, and this bijection gives a combinatorial model for the decomposition of the k-th tensor power of the symmetric group permutation module into irreducibles. The algorithm embeds the crystal graphs of these irreducible modules into k-dimensional Euclidean space at the point corresponding to its given word, and we explore connections between the algebra and the geometry of this embedding.
Bass, Connor, "Crystal Graphs for the Symmetric Group" (2022). Mathematics, Statistics, and Computer Science Honors Projects. 66.
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