10,013 research outputs found
Small-span Hermitian matrices over quadratic integer rings
Building on the classification of all characteristic polynomials of integer
symmetric matrices having small span (span less than 4), we obtain a
classification of small-span polynomials that are the characteristic polynomial
of a Hermitian matrix over some quadratic integer ring. Taking quadratic
integer rings as our base, we obtain as characteristic polynomials some
low-degree small-span polynomials that are not the characteristic (or minimal)
polynomial of any integer symmetric matrix.Comment: 16 page
Small ball probability, Inverse theorems, and applications
Let be a real random variable with mean zero and variance one and
be a multi-set in . The random sum
where are iid copies of
is of fundamental importance in probability and its applications.
We discuss the small ball problem, the aim of which is to estimate the
maximum probability that belongs to a ball with given small radius,
following the discovery made by Littlewood-Offord and Erdos almost 70 years
ago. We will mainly focus on recent developments that characterize the
structure of those sets where the small ball probability is relatively
large. Applications of these results include full solutions or significant
progresses of many open problems in different areas.Comment: 47 page
Combinatorial methods of character enumeration for the unitriangular group
Let \UT_n(q) denote the group of unipotent upper triangular
matrices over a field with elements. The degrees of the complex irreducible
characters of \UT_n(q) are precisely the integers with , and it has been
conjectured that the number of irreducible characters of \UT_n(q) with degree
is a polynomial in with nonnegative integer coefficients (depending
on and ). We confirm this conjecture when and is arbitrary
by a computer calculation. In particular, we describe an algorithm which allows
us to derive explicit bivariate polynomials in and giving the number of
irreducible characters of \UT_n(q) with degree when and . When divided by and written in terms of the variables
and , these functions are actually bivariate polynomials with nonnegative
integer coefficients, suggesting an even stronger conjecture concerning such
character counts. As an application of these calculations, we are able to show
that all irreducible characters of \UT_n(q) with degree are
Kirillov functions. We also discuss some related results concerning the problem
of counting the irreducible constituents of individual supercharacters of
\UT_n(q).Comment: 34 pages, 5 table
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