2025/07/03 by Campbell, John M. · 1 citation
#05E10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.02539
Given a direct sum A of full matrix algebras, if there is a combinatorial interpretation associated with both the dimension of A and the dimensions of the irreducible A-modules, then this can be thought of as providing an analogue of the famous Frobenius-Young identity n! = ∑λ\vdash n ( fλ )2 derived from the semisimple structure of the symmetric group algebra ℂSn, letting fλ denote the number of Young tableaux of partition shape λ\vdash n. By letting gα denote the number of standard immaculate tableaux of composition shape α\vDash n, we construct an algebra ℂIn with a semisimple structure such that dim ℂIn = ∑α\vDash n (gα)2 and such that ℂIn contains an isomorphic copy of ℂSn. We bijectively prove a recurrence for dim ℂIn so as to construct a basis of ℂIn indexed by permutation-like objects that we refer to as immacutations. We form a basis Bn of ℂIn such that ℂ Bn has the structure of a monoid algebra in such a way so that Bn is closed under the multiplicative operation of ℂ In, yielding a monoid structure on the set of order-n immacutations.