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Constructing equivariant vector bundles via the BGG correspondence
We describe a strategy for the construction of finitely generated
-equivariant -graded modules over the exterior algebra for a
finite group . By an equivariant version of the BGG correspondence,
defines an object in the bounded derived category of
-equivariant coherent sheaves on projective space. We develop a necessary
condition for being isomorphic to a vector bundle that can be
simply read off from the Hilbert series of . Combining this necessary
condition with the computation of finite excerpts of the cohomology table of
makes it possible to enlist a class of equivariant vector bundles
on that we call strongly determined in the case where is the
alternating group on points
Quantum supergroups and topological invariants of three - manifolds
The Reshetikhin - Turaeve approach to topological invariants of three -
manifolds is generalized to quantum supergroups. A general method for
constructing three - manifold invariants is developed, which requires only the
study of the eigenvalues of certain central elements of the quantum supergroup
in irreducible representations. To illustrate how the method works,
at odd roots of unity is studied in detail, and the
corresponding topological invariants are obtained.Comment: 22 page
Beating the Generator-Enumeration Bound for -Group Isomorphism
We consider the group isomorphism problem: given two finite groups G and H
specified by their multiplication tables, decide if G cong H. For several
decades, the n^(log_p n + O(1)) generator-enumeration bound (where p is the
smallest prime dividing the order of the group) has been the best worst-case
result for general groups. In this work, we show the first improvement over the
generator-enumeration bound for p-groups, which are believed to be the hard
case of the group isomorphism problem. We start by giving a Turing reduction
from group isomorphism to n^((1 / 2) log_p n + O(1)) instances of p-group
composition-series isomorphism. By showing a Karp reduction from p-group
composition-series isomorphism to testing isomorphism of graphs of degree at
most p + O(1) and applying algorithms for testing isomorphism of graphs of
bounded degree, we obtain an n^(O(p)) time algorithm for p-group
composition-series isomorphism. Combining these two results yields an algorithm
for p-group isomorphism that takes at most n^((1 / 2) log_p n + O(p)) time.
This algorithm is faster than generator-enumeration when p is small and slower
when p is large. Choosing the faster algorithm based on p and n yields an upper
bound of n^((1 / 2 + o(1)) log n) for p-group isomorphism.Comment: 15 pages. This is an updated and improved version of the results for
p-groups in arXiv:1205.0642 and TR11-052 in ECC
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