The characteristic polynomial of the Adams operators on graded connected Hopf algebras

Abstract

The Adams operators Ψ n on a Hopf algebra H are the convolution powers of the identity of H. They are also called Hopf powers or Sweedler powers. We study the Adams operators when H is graded connected. The main result is a complete description of the characteristic polynomial-both eigenvalues and their multiplicities-for the action of the operator Ψ n on each homogeneous component of H. The eigenvalues are powers of n. The multiplicities are independent of n, and in fact only depend on the dimension sequence of H. These results apply in particular to the antipode of H, as the case n = −1. We obtain closed forms for the generating function of the sequence of traces of the Adams operators. In the case of the antipode, the generating function bears a particularly simple relationship to the one for the dimension sequence. In case H is cofree, we give an alternative description for the characteristic polynomial and the trace of the antipode in terms of certain palindromic words. We discuss parallel results that hold for Hopf monoids in species and for q-Hopf algebras.

On the square of the antipode in a connected filtered Hopf algebra

“…The equality (34) in Corollary 2.17 generalizes [AguLau14, Example 8]. Indeed, if H is the Malvenuto-Reutenauer Hopf algebra 2 , then the condition (31) is satisfied (since H 1 is a free k-module of rank 1 in this case); therefore, Corollary 2.17 (c) can be applied in this case, and we recover [AguLau14,Example 8]. Likewise, we can obtain the same result if H is the Hopf algebra WQSym of word quasisymmetric functions 3 .…”

“…For specific combinatorially interesting Hopf algebras, even stronger results hold; in particular, id −S 2 n−1 (H n ) = 0 holds for each n > 1 when H is the Malvenuto-Reutenauer Hopf algebra (see [AguLau14,Example 8]).…”

The Eigenvalues of Hyperoctahedral Descent Operators and Applications to Card-Shuffling

“…iii) As noted in [AL15], τ and gr(τ ) have the same spectrum, and so does (gr τ ) * . Since τ and (gr τ ) * are involutions, they are diagonalisable, and so their spectrum determines their eigenspace dimensions.…”

“…4.5] for their connections to Hochschild homology. Their eigenvalues and multiplicities are obtained in [AL15], and applied to derive some combinatorial identities. The paper [DPR14] gives a basis of eigenvectors for m [a] •∆ [a] on free-commutative or cocommutative algebras, and interprets the matrix of these Adams operations as the transition probabilities of a Markov chain on a basis of H. When H is the shuffle algebra, this probabilistic interpretation recovers the famous Gilbert-Shannon-Reeds riffle-shuffle of a deck of cards [BD92]: cut the deck into a piles according to the multinomial distribution, then interleave the piles together so cards from the same pile stay in the same relative order.…”

Polyadic Hopf Algebras and Quantum Groups

“…Remark 91. If the polyadic tensor product and the underlying polyadic field k are derived (see discussion [10]), while all maps coincide f (i) = f, the convolution product (170) is called the Sweedler power of f [34] or the Adams operator [35]. In the binary case they denoted it by (f) n , but for the n -ary product this is the first polyadic power of f (see (6)).…”