96,576 research outputs found
Fourier-Reflexive Partitions and MacWilliams Identities for Additive Codes
A partition of a finite abelian group gives rise to a dual partition on the
character group via the Fourier transform. Properties of the dual partitions
are investigated and a convenient test is given for the case that the bidual
partition coincides the primal partition. Such partitions permit MacWilliams
identities for the partition enumerators of additive codes. It is shown that
dualization commutes with taking products and symmetrized products of
partitions on cartesian powers of the given group. After translating the
results to Frobenius rings, which are identified with their character module,
the approach is applied to partitions that arise from poset structures
Schur Positivity and Kirillov-Reshetikhin Modules
In this note, inspired by the proof of the Kirillov-Reshetikhin conjecture,
we consider tensor products of Kirillov-Reshetikhin modules of a fixed node and
various level. We fix a positive integer and attach to each of its partitions
such a tensor product. We show that there exists an embedding of the tensor
products, with respect to the classical structure, along with the reverse
dominance relation on the set of partitions
Integrality of hook ratios
We study integral ratios of hook products of quotient partitions. This
question is motivated by an analogous question in number theory concerning
integral factorial ratios. We prove an analogue of a theorem of Landau that
already applied in the factorial case. Under the additional condition that the
ratio has one more factor on the denominator than the numerator, we provide a
complete classification. Ultimately this relies on Kneser's theorem in additive
combinatorics.Comment: 13 pages, 3 figures Keywords: partitions, hook products, Kneser's
theorem, McKay numbers, Beurling-Nyman criterio
Reduced Kronecker products which are multiplicity free or contain only few components
It is known that the Kronecker coefficient of three partitions is a bounded
and weakly increasing sequence if one increases the first part of all three
partitions. Furthermore if the first parts of partitions \lambda,\mu are big
enough then the coefficients of the Kronecker product [\lambda][\mu]=\sum_\n
g(\l,\m,\n)[\nu] do not depend on the first part but only on the other parts.
The reduced Kronecker product [\lambda]_\bullet \star[\mu]_\bullet can be
viewed (roughly) as the Kronecker product [(n-|\lambda|,\lambda)][(n-|\mu|,\m)]
for n big enough. In this paper we classify the reduced Kronecker products
which are multiplicity free and those which contain less than 10 components.We
furthermore give general lower bounds for the number of constituents and
components of a given reduced Kronecker product. We also give a lower bound for
the number of pairs of components whose corresponding partitions differ by one
box. Finally we argue that equality of two reduced Kronecker products is only
possible in the trivial case that the factors of the product are the same.Comment: 11 pages, final version. appears in European J. Combi
Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions
We show that, for a certain class of partitions and an even number of
variables of which half are reciprocals of the other half, Schur polynomials
can be factorized into products of odd and even orthogonal characters. We also
obtain related factorizations involving sums of two Schur polynomials, and
certain odd-sized sets of variables. Our results generalize the factorization
identities proved by Ciucu and Krattenthaler (Advances in combinatorial
mathematics, 39-59, 2009) for partitions of rectangular shape. We observe that
if, in some of the results, the partitions are taken to have rectangular or
double-staircase shapes and all of the variables are set to 1, then
factorization identities for numbers of certain plane partitions, alternating
sign matrices and related combinatorial objects are obtained.Comment: 22 pages; v2: minor changes, published versio
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