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The characteristic polynomial of the Adams operators on graded connected Hopf algebras
The Adams operators on a Hopf algebra are the convolution powers
of the identity of . We study the Adams operators when is graded
connected. They are also called Hopf powers or Sweedler powers. The main result
is a complete description of the characteristic polynomial (both eigenvalues
and their multiplicities) for the action of the operator on each
homogeneous component of . The eigenvalues are powers of . The
multiplicities are independent of , and in fact only depend on the dimension
sequence of . These results apply in particular to the antipode of (the
case ). 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 -Hopf algebras.Comment: 36 pages; two appendice
Combinatorial Nullstellensatz modulo prime powers and the Parity Argument
We present new generalizations of Olson's theorem and of a consequence of
Alon's Combinatorial Nullstellensatz. These enable us to extend some of their
combinatorial applications with conditions modulo primes to conditions modulo
prime powers. We analyze computational search problems corresponding to these
kinds of combinatorial questions and we prove that the problem of finding
degree-constrained subgraphs modulo such as -divisible subgraphs and
the search problem corresponding to the Combinatorial Nullstellensatz over
belong to the complexity class Polynomial Parity Argument (PPA)
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