21,680 research outputs found
Quantum invariant families of matrices in free probability
We consider (self-adjoint) families of infinite matrices of noncommutative
random variables such that the joint distribution of their entries is invariant
under conjugation by a free quantum group. For the free orthogonal and
hyperoctahedral groups, we obtain complete characterizations of the invariant
families in terms of an operator-valued -cyclicity condition. This is a
surprising contrast with the Aldous-Hoover characterization of jointly
exchangeable arrays.Comment: 33 page
Finite Affine Groups: Cycle Indices, Hall-Littlewood Polynomials, and Probabilistic Algorithms
The asymptotic study of the conjugacy classes of a random element of the
finite affine group leads one to define a probability measure on the set of all
partitions of all positive integers. Four different probabilistic
understandings of this measure are given--three using symmetric function theory
and one using Markov chains. This leads to non-trivial enumerative results.
Cycle index generating functions are derived and are used to compute the large
dimension limiting probabilities that an element of the affine group is
separable, cyclic, or semisimple and to study the convergence to these limits.
This yields the first examples of such computations for a maximal parabolic
subgroup of a finite classical group.Comment: Revised version, to appear in J. Algebra. A few typos are fixed; no
substantive change
The cyclic sieving phenomenon: a survey
The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a
2004 paper. Let X be a finite set, C be a finite cyclic group acting on X, and
f(q) be a polynomial in q with nonnegative integer coefficients. Then the
triple (X,C,f(q)) exhibits the cyclic sieving phenomenon if, for all g in C, we
have # X^g = f(w) where # denotes cardinality, X^g is the fixed point set of g,
and w is a root of unity chosen to have the same order as g. It might seem
improbable that substituting a root of unity into a polynomial with integer
coefficients would have an enumerative meaning. But many instances of the
cyclic sieving phenomenon have now been found. Furthermore, the proofs that
this phenomenon hold often involve interesting and sometimes deep results from
representation theory. We will survey the current literature on cyclic sieving,
providing the necessary background about representations, Coxeter groups, and
other algebraic aspects as needed.Comment: 48 pages, 3 figures, the sedcond version contains numerous changes
suggested by colleagues and the referee. To appear in the London Mathematical
Society Lecture Note Series. The third version has a few smaller change
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