8,579 research outputs found
A gluing construction for polynomial invariants
We give a polynomial gluing construction of two groups and which results in a group
whose ring of invariants is isomorphic to the
tensor product of the rings of invariants of and . In particular,
this result allows us to obtain many groups with polynomial rings of
invariants, including all -groups whose rings of invariants are polynomial
over , and the finite subgroups of defined by
sparsity patterns, which generalize many known examples.Comment: 10 pages, to appear in Journal of algebr
MacWilliams' Extension Theorem for Bi-Invariant Weights over Finite Principal Ideal Rings
A finite ring R and a weight w on R satisfy the Extension Property if every
R-linear w-isometry between two R-linear codes in R^n extends to a monomial
transformation of R^n that preserves w. MacWilliams proved that finite fields
with the Hamming weight satisfy the Extension Property. It is known that finite
Frobenius rings with either the Hamming weight or the homogeneous weight
satisfy the Extension Property. Conversely, if a finite ring with the Hamming
or homogeneous weight satisfies the Extension Property, then the ring is
Frobenius.
This paper addresses the question of a characterization of all bi-invariant
weights on a finite ring that satisfy the Extension Property. Having solved
this question in previous papers for all direct products of finite chain rings
and for matrix rings, we have now arrived at a characterization of these
weights for finite principal ideal rings, which form a large subclass of the
finite Frobenius rings. We do not assume commutativity of the rings in
question.Comment: 12 page
Clifford correspondence for algebras
We give a Clifford correspondence for an algebra A over an algebraically
closed field, that is an algorithm for constructing some finite-dimensional
simple A-modules from simple modules for a subalgebra and endomorphism
algebras. This applies to all finite-dimensional simple A-modules in the case
that A is finite-dimensional and semisimple with a given semisimple subalgebra.
We discuss connections between our work and earlier results, with a view
towards applications particularly to finite-dimensional semisimple Hopf
algebras.Comment: 12 page
Yang-Mills theory and Tamagawa numbers
Atiyah and Bott used equivariant Morse theory applied to the Yang-Mills
functional to calculate the Betti numbers of moduli spaces of vector bundles
over a Riemann surface, rederiving inductive formulae obtained from an
arithmetic approach which involved the Tamagawa number of SL_n. This article
surveys this link between Yang-Mills theory and Tamagawa numbers, and explains
how methods used over the last three decades to study the singular cohomology
of moduli spaces of bundles on a smooth complex projective curve can be adapted
to the setting of A^1-homotopy theory to study the motivic cohomology of these
moduli spaces.Comment: Accepted for publication in the Bulletin of the London Mathematical
Societ
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