Fast Fourier transforms for the rook monoid

Fast Fourier transforms for the rook monoid (0)

by M Malandro, D Rockmore

Venue:

Trans. Amer. Math. Soc

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Fast Fourier transforms for finite inverse semigroups

by Martin E. Malandro - J. Algebra

"... Abstract. We extend the theory of fast Fourier transforms on finite groups to finite inverse semigroups. We use a general method for constructing the irreducible representations of a finite inverse semigroup to reduce the problem of computing its Fourier transform to the problems of computing Fourie ..."

Abstract - Cited by 4 (2 self) - Add to MetaCart

Abstract. We extend the theory of fast Fourier transforms on finite groups to finite inverse semigroups. We use a general method for constructing the irreducible representations of a finite inverse semigroup to reduce the problem of computing its Fourier transform to the problems of computing Fourier transforms on its maximal subgroups and a fast zeta transform on its poset structure. We then exhibit explicit fast algorithms for particular inverse semigroups of interest—specifically, for the rook monoid and its wreath products by arbitrary finite groups. 1.

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Fast Zeta Transforms for Lattices with Few Irreducibles

by Andreas Björklund, Thore Husfeldt, Petteri Kaski, Mikko Koivisto, Jesper Nederlof, Pekka Parviainen , 2012

"... We investigate fast algorithms for changing between the standard basis and an orthogonal basis of idempotents for Möbius algebras of finite lattices. We show that every lattice with v elements, n of which are nonzero and join-irreducible (or, by a dual result, nonzero and meet-irreducible), has arit ..."

Abstract - Cited by 3 (1 self) - Add to MetaCart

We investigate fast algorithms for changing between the standard basis and an orthogonal basis of idempotents for Möbius algebras of finite lattices. We show that every lattice with v elements, n of which are nonzero and join-irreducible (or, by a dual result, nonzero and meet-irreducible), has arithmetic circuits of size O(vn) for computing the zeta transform and its inverse, thus enabling fast multiplication in the Möbius algebra. Furthermore, the circuit construction in fact gives optimal (up to constants) circuits for a number of lattices of combinatorial and algebraic relevance, such as the lattice of subsets of a finite set, the lattice of set partitions of a finite set, the lattice of vector subspaces of a finite vector space, and the lattice of positive divisors of a positive integer.

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