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We show that the Nakayama automorphism of a Frobenius algebra $R$ over a field $k$ is independent of the field (Theorem 4). Consequently, the $k$-dual functor on left $R$-modules and the bimodule isomorphism type of the $k$-dual of $R$, and hence the question of whether $R$ is a symmetric $k$-algebra, are independent of $k$. We give a purely ring-theoretic condition that is necessary and sufficient for a finite-dimensional algebra over an infinite field to be a symmetric algebra (Theorem 7). Key words: Nakayama automorphism, Frobenius algebra, Frobenius ring, symmetric algebra, dual module, dual functor, bimodule, Brauer Equivalence.Comment: 11 pages, 1 figur

We analyze the homothety types of associative bilinear forms that can occur on a Hopf algebra or on a local Frobenius k-algebra R with residue field k. If R is symmetric, then there exists a unique form on R up to homothety iff R is commutative. If R is Frobenius, then we introduce a norm based on the Nakayama automorphism of R. We show that if two forms on R are homothetic, then the norm of the unit separating them is central, and we conjecture the converse. We show that if the dimension of R is even, then the determinant of a form on R, taken ink/k 2 , is an invariant for R.  2005 Elsevier Inc. All rights reserved.

Consider a system of \(n\) players in which each initially starts on a different team. At each time step, we select an individual winner and an individual loser randomly and the loser joins the winner's team. The resulting Markov chain and stochastic matrix clearly have one absorbing state, in which all players are on the same team, but the combinatorics along the way are surprisingly elegant. The expected number of time steps until each team is eliminated is a ratio of binomial coefficients. When a team is eliminated, the probabilities that the players are configured in various partitions of \(n\) into \(t\) teams are given by multinomial coefficients. The expected value of the time to absorbtion is \((n-1)^2\) steps. The results depend on elementary combinatorics, linear algebra, and the theory of Markov chains

Benjamin and Robert Mena for suggesting this investigation and many useful subsequent thoughts. The referees also offered many helpful ideas that dramatically improved this paper.