Representations of Some Hopf Algebras Associated to the Symmetric Group S n

Abstract

We prove that all irreducible representations of the bismash product H n = k Cn #kS n−1 have Frobenius-Schur indicator +1, where k is an algebraically closed field of characteristic 0. If n = p, a prime, we find all indicators for J n = k Sn−1 #k Cn . We also study more general bismash products.

Some bismash products that are not group algebras

“…Then G = SC and S has exactly two orbits in its action on C with representatives 1 and π and respective stabilisers Stab S (1 C ) = S and (where we multiply permutations from left to right) Stab S (π) = Stab G (n − 1, n). As has also been observed in [3], Proposition 1 now immediately yields the following. We will also dispose of a simple number-theoretic fact here.…”

“…By Lemma 3, this is an immediate contradiction for q odd. For q = 2, we note that the index [G : O p (G)] is divisible by 8, appealing to the solubility of groups of odd order, 3 and Lemma 3 again yields a contradiction.…”

“…Susan Montgomery and others [3,5] have recently studied a family of finite-dimensional complex semisimple Hopf algebras which are constructed as so-called bismash products from the factorisations of symmetric groups. She and Jacques Alev have then asked whether these algebras have the same algebra structure as some group algebra.…”

Frobenius-Schur Indicators for Some Fusion Categories Associated to Symmetric and Alternating Groups

“…(In the presence of cohomological data, one can similarly obtain bicross product Hopf algebras.) For such bismash product Hopf algebras, many results on indicator values were previously obtained (and some questions asked) by other authors [3,2,8], and these results have provided part of the motivation for the present paper. As it turns out, the indicator formulas for C(G, H) are not only applicable in more general situations (where a direct factorization is not available), but they also offer a significant advantage in the special case of a bismash product.…”

Indicators of bismash products from exact symmetric group factorizations

“…Most of our compuations will be done using [GAP4] with custom programs. Our main result is an extension of [JM,Theorem 3.6], which stated the Hopf algebra H n = k Cn #kS n−1 arising from the factorization of S n = S n−1 · C n with C n = (1, 2, . .…”