57,076 research outputs found
Fusion multiplicities as polytope volumes: N-point and higher-genus su(2) fusion
We present the first polytope volume formulas for the multiplicities of
affine fusion, the fusion in Wess-Zumino-Witten conformal field theories, for
example. Thus, we characterise fusion multiplicities as discretised volumes of
certain convex polytopes, and write them explicitly as multiple sums measuring
those volumes. We focus on su(2), but discuss higher-point (N>3) and
higher-genus fusion in a general way. The method follows that of our previous
work on tensor product multiplicities, and so is based on the concepts of
generalised Berenstein-Zelevinsky diagrams, and virtual couplings. As a
by-product, we also determine necessary and sufficient conditions for
non-vanishing higher-point fusion multiplicities. In the limit of large level,
these inequalities reduce to very simple non-vanishing conditions for the
corresponding tensor product multiplicities. Finally, we find the minimum level
at which the higher-point fusion and tensor product multiplicities coincide.Comment: 14 pages, LaTeX, version to be publishe
The F-signature of an affine semigroup ring
We prove that the F-signature of an affine semigroup ring of positive
characteristic is always a rational number, and describe a method for computing
this number. We use this method to determine the F-signature of Segre products
of polynomial rings, and of Veronese subrings of polynomial rings. Our
technique involves expressing the F-signature of an affine semigroup ring as
the difference of the Hilbert-Kunz multiplicities of two monomial ideals, and
then using Watanabe's result that these Hilbert-Kunz multiplicities are
rational numbers
On the Imaginary Simple Roots of the Borcherds Algebra
In a recent paper (hep-th/9703084) it was conjectured that the imaginary
simple roots of the Borcherds algebra at level 1 are its only
ones. We here propose an independent test of this conjecture, establishing its
validity for all roots of norm . However, the conjecture fails for
roots of norm -10 and beyond, as we show by computing the simple multiplicities
down to norm -24, which turn out to be remakably small in comparison with the
corresponding multiplicities. Our derivation is based on a modified
denominator formula combining the denominator formulas for and
, and provides an efficient method for determining the imaginary
simple roots. In addition, we compute the multiplicities of all roots
up to height 231, including levels up to and norms -42.Comment: 14 pages, LaTeX2e, packages amsmath, amsfonts, amssymb, amsthm,
xspace, pstricks, longtable; substantially extended, appendix with new
root multiplicities adde
Multiplicity Preserving Triangular Set Decomposition of Two Polynomials
In this paper, a multiplicity preserving triangular set decomposition
algorithm is proposed for a system of two polynomials. The algorithm decomposes
the variety defined by the polynomial system into unmixed components
represented by triangular sets, which may have negative multiplicities. In the
bivariate case, we give a complete algorithm to decompose the system into
multiplicity preserving triangular sets with positive multiplicities. We also
analyze the complexity of the algorithm in the bivariate case. We implement our
algorithm and show the effectiveness of the method with extensive experiments.Comment: 18 page
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