36,493 research outputs found
Graham Higman's PORC theorem
Graham Higman published two important papers in 1960. In the first of these
papers he proved that for any positive integer the number of groups of
order is bounded by a polynomial in , and he formulated his famous
PORC conjecture about the form of the function giving the number of
groups of order . In the second of these two papers he proved that the
function giving the number of -class two groups of order is PORC. He
established this result as a corollary to a very general result about vector
spaces acted on by the general linear group. This theorem takes over a page to
state, and is so general that it is hard to see what is going on. Higman's
proof of this general theorem contains several new ideas and is quite hard to
follow. However in the last few years several authors have developed and
implemented algorithms for computing Higman's PORC formulae in special cases of
his general theorem. These algorithms give perspective on what are the key
points in Higman's proof, and also simplify parts of the proof.
In this note I give a proof of Higman's general theorem written in the light
of these recent developments
On conformally covariant powers of the Laplacian
We propose and discuss recursive formulas for conformally covariant powers
of the Laplacian (GJMS-operators). For locally conformally flat
metrics, these describe the non-constant part of any GJMS-operator as the sum
of a certain linear combination of compositions of lower order GJMS-operators
(primary part) and a second-order operator which is defined by the Schouten
tensor (secondary part). We complete the description of GJMS-operators by
proposing and discussing recursive formulas for their constant terms, i.e., for
Branson's -curvatures, along similar lines. We confirm the picture in a
number of cases. Full proofs are given for spheres of any dimension and
arbitrary signature. Moreover, we prove formulas of the respective critical
third power in terms of the Yamabe operator and the Paneitz
operator , and of a fourth power in terms of , and . For
general metrics, the latter involves the first two of Graham's extended
obstruction tensors. In full generality, the recursive formulas remain
conjectural. We describe their relation to the theory of residue families and
the associated -curvature polynomials.Comment: We extend the previous description of GJMS-operators to general
metrics (Conjecture 11.1
Centres of Hecke algebras: the Dipper-James conjecture
In this paper we prove the Dipper-James conjecture that the centre of the
Iwahori-Hecke algebra of type A is the set of symmetric polynomials in the
Jucys-Murphy operators.Comment: 27 pages. To appear J. Algebr
An approximate version of Sidorenko's conjecture
A beautiful conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if H
is a bipartite graph, then the random graph with edge density p has in
expectation asymptotically the minimum number of copies of H over all graphs of
the same order and edge density. This conjecture also has an equivalent
analytic form and has connections to a broad range of topics, such as matrix
theory, Markov chains, graph limits, and quasirandomness. Here we prove the
conjecture if H has a vertex complete to the other part, and deduce an
approximate version of the conjecture for all H. Furthermore, for a large class
of bipartite graphs, we prove a stronger stability result which answers a
question of Chung, Graham, and Wilson on quasirandomness for these graphs.Comment: 12 page
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