244 research outputs found
Symmetry-based matrix factorization
AbstractWe present a method for factoring a given matrix M into a short product of sparse matrices, provided that M has a suitable “symmetry”. This sparse factorization represents a fast algorithm for the matrix–vector multiplication with M. The factorization method consists of two essential steps. First, a combinatorial search is used to compute a suitable symmetry of M in the form of a pair of group representations. Second, the group representations are decomposed stepwise, which yields factorized decomposition matrices and determines a sparse factorization of M. The focus of this article is the first step, finding the symmetries. All algorithms described have been implemented in the library AREP. We present examples for automatically generated sparse factorizations—and hence fast algorithms—for a class of matrices corresponding to digital signal processing transforms including the discrete Fourier, cosine, Hartley, and Haar transforms
Extended -System of Type
We prove a family of 3-term relations in the Grothendieck ring of the
category of finite-dimensional modules over the affine quantum algebra of type
extending the celebrated -system relations of type . We show that
these relations can be used to compute classes of certain irreducible modules,
including classes of all minimal affinizations of type . We use this
result to obtain explicit formulas for dimensions of all participating modules
Path description of type B q-characters
We give a set of sufficient conditions for a Laurent polynomial to be the
q-character of a finite-dimensional irreducible representation of a quantum
affine group. We use this result to obtain an explicit path description of
q-characters for a class of modules in type B. In particular, this proves a
conjecture of Kuniba-Ohta-Suzuki.Comment: 32 pages, late
Extended T-systems
We use the theory of q-characters to establish a number of short exact
sequences in the category of finite-dimensional representations of the quantum
affine groups of types A and B. That allows us to introduce a set of 3-term
recurrence relations which contains the celebrated T-system as a special case.Comment: 36 pages, latex; v2: version to appear in Selecta Mathematic
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