87,114 research outputs found
Calabi-Yau manifolds and sporadic groups
A few years ago a connection between the elliptic genus of the K3 manifold
and the largest Mathieu group M was proposed. We study the elliptic
genera for Calabi-Yau manifolds of larger dimensions and discuss potential
connections between the expansion coefficients of these elliptic genera and
sporadic groups. While the Calabi-Yau 3-fold case is rather uninteresting, the
elliptic genera of certain Calabi-Yau -folds for have expansions that
could potentially arise from underlying sporadic symmetry groups. We explore
such potential connections by calculating twined elliptic genera for a large
number of Calabi-Yau 5-folds that are hypersurfaces in weighted projected
spaces, for a toroidal orbifold and two Gepner models.Comment: 34 pages;v2 minor correction
The generic character table of a Sylow -subgroup of a finite Chevalley group of type
Let be a Sylow -subgroup of the finite Chevalley group of type
over the field of elements, where is a power of a prime . We
describe a construction of the generic character table of
Calculating conjugacy classes in Sylow p-subgroups of finite Chevalley groups of rank six and seven
Let G(q) be a finite Chevalley group, where q is a power of a good prime p,
and let U(q) be a Sylow p-subgroup of G(q). Then a generalized version of a
conjecture of Higman asserts that the number k(U(q)) of conjugacy classes in
U(q) is given by a polynomial in q with integer coefficients. In an earlier
paper, the first and the third authors developed an algorithm to calculate the
values of k(U(q)). By implementing it into a computer program using GAP, they
were able to calculate k(U(q)) for G of rank at most 5, thereby proving that
for these cases k(U(q)) is given by a polynomial in q. In this paper we present
some refinements and improvements of the algorithm that allow us to calculate
the values of k(U(q)) for finite Chevalley groups of rank six and seven, except
E_7. We observe that k(U(q)) is a polynomial, so that the generalized Higman
conjecture holds for these groups. Moreover, if we write k(U(q)) as a
polynomial in q-1, then the coefficients are non-negative.
Under the assumption that k(U(q)) is a polynomial in q-1, we also give an
explicit formula for the coefficients of k(U(q)) of degrees zero, one and two.Comment: 16 page
Asymptotics of characters of symmetric groups related to Stanley character formula
We prove an upper bound for characters of the symmetric groups. Namely, we
show that there exists a constant a>0 with a property that for every Young
diagram \lambda with n boxes, r(\lambda) rows and c(\lambda) columns |Tr
\rho^\lambda(\pi) / Tr \rho^\lambda(e)| < [a max(r(\lambda)/n,
c(\lambda)/n,|\pi|/n) ]^{|\pi|}, where |\pi| is the minimal number of factors
needed to write \pi\in S_n as a product of transpositions. We also give uniform
estimates for the error term in the Vershik-Kerov's and Biane's character
formulas and give a new formula for free cumulants of the transition measure.Comment: Version 4: Change of title, shortened to 20 pages. Version 3: 24
pages, the title and the list of authors were changed. Version 2: 14 pages,
the title, abstract and the main result were changed. Version 1: 10 pages
(mistake in Lemma 7- which is false
Irreducible representations of the rational Cherednik algebra associated to the Coxeter group H_3
This paper describes irreducible representations in category O of the
rational Cherednik algebra H_c(H_3,h) associated to the exceptional Coxeter
group H_3 and any complex parameter c. We compute the characters of all these
representations explicitly. As a consequence, we classify all the finite
dimensional irreducible representations of H_c(H_3,h).Comment: Version 2: a minor error in 5.1 and its consequences in 3.1. and 3.2
corrected. Comments welcom
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