3,089 research outputs found
Cylindric Reverse Plane Partitions and 2D TQFT
The ring of symmetric functions carries the structure of a Hopf algebra. When
computing the coproduct of complete symmetric functions one arrives
at weighted sums over reverse plane partitions (RPP) involving binomial
coefficients. Employing the action of the extended affine symmetric group at
fixed level we generalise these weighted sums to cylindric RPP and define
cylindric complete symmetric functions. The latter are shown to be
-positive, that is, their expansions coefficients in the basis of complete
symmetric functions are non-negative integers. We state an explicit formula in
terms of tensor multiplicities for irreducible representations of the
generalised symmetric group. Moreover, we relate the cylindric complete
symmetric functions to a 2D topological quantum field theory (TQFT) that is a
generalisation of the celebrated -Verlinde algebra
or Wess-Zumino-Witten fusion ring, which plays a prominent role in the context
of vertex operator algebras and algebraic geometry.Comment: 13 pages, 1 figure, accepted conference proceedings article for
FPSAC2018 (Hanover
Families of generalized Kloosterman sums
We construct p-adic relative cohomology for a family of toric exponential
sums which generalize the classical Kloosterman sums. Under natural hypotheses
such as quasi-homogeneity and nondegeneracy, this cohomology is acyclic except
in the top dimension. Our construction enables sufficiently sharp estimates for
the action of Frobenius on cohomology so that our earlier work may be applied
to the L-functions coming from linear algebra operations on these families to
deduce a number of basic properties.Comment: 36 pages, 4 figure
Galois cohomology of a number field is Koszul
We prove that the Milnor ring of any (one-dimensional) local or global field
K modulo a prime number l is a Koszul algebra over Z/l. Under mild assumptions
that are only needed in the case l=2, we also prove various module Koszulity
properties of this algebra. This provides evidence in support of Koszulity
conjectures that were proposed in our previous papers. The proofs are based on
the Class Field Theory and computations with quadratic commutative Groebner
bases (commutative PBW-bases).Comment: LaTeX 2e, 25 pages; v.2: minor grammatic changes; v.3: classical
references added, remark inserted in subsection 1.6, details of arguments
added in subsections 1.4, 1.7 and sections 5 and 6; v.4: still more misprints
corrected, acknowledgement updated, a sentence inserted in section 4, a
reference added -- this is intended as the final versio
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