5,794 research outputs found
Beyond Rouquier partitions
We obtain closed formulas, in terms of Littlewood-Richardson coefficients,
for the canonical basis elements of the Fock space representation of
which are labelled by partitions having 'locally
small' -quotients and arbitrary -cores. We further show that, upon
evaluation at , this gives the corresponding decomposition numbers of the
-Schur algebras in characteristic (where is a primitive -th root
of unity if and otherwise) whenever is greater than the
size of each constituent of the -quotient.Comment: 17 pages. This replaces the earlier version entitled 'Some
decomposition numbers of q-Shcur algebra
Parities of v-decomposition numbers and an application to symmetric group algebras
We prove that the v-decomposition number is an even or
odd polynomial according to whether the partitions and have the
same relative sign (or parity) or not. We then use this result to verify
Martin's conjecture for weight 3 blocks of symmetric group algebras -- that
these blocks have the property that their projective (indecomposable) modules
have a common radical length 7.Comment: 21 page
The non-projective part of the Lie module for the symmetric group
The Lie module of the group algebra of the symmetric group is known to
be not projective if and only if the characteristic of divides . We
show that in this case its non-projective summands belong to the principal
block of .
Let be a vector space of dimension over , and let be the
-th homogeneous part of the free Lie algebra on ; this is a polynomial
representation of of degree , or equivalently, a module of the
Schur algebra . Our result implies that, when , every summand
of which is not a tilting module belongs to the principal block of
, by which we mean the block containing the -th symmetric power of
The Lie module of the symmetric group
We provide an upper bound for the dimension of the maximal projective
submodule of the Lie module of the symmetric group of letters in prime
characteristic , where with .Comment: 18 page
Homomorphisms from Specht Modules to Signed Young Permutation Modules
We construct a class of homomorphisms from a Specht
module to a signed permutation module
which generalises James's construction of
homomorphisms whose codomain is a Young permutation module. We show that any
lies in the -span of
, a subset of corresponding to
semistandard -tableaux of type . We also study the
conditions for which - a subset of
induced by - is linearly independent, and show that it
is a basis for
when is semisimple
The study of a new gerrymandering methodology
This paper is to obtain a simple dividing-diagram of the congressional
districts, where the only limit is that each district should contain the same
population if possibly. In order to solve this problem, we introduce three
different standards of the "simple" shape. The first standard is that the final
shape of the congressional districts should be of a simplest figure and we
apply a modified "shortest split line algorithm" where the factor of the same
population is considered only. The second standard is that the gerrymandering
should ensure the integrity of the current administrative area as the
convenience for management. Thus we combine the factor of the administrative
area with the first standard, and generate an improved model resulting in the
new diagram in which the perimeters of the districts are along the boundaries
of some current counties. Moreover, the gerrymandering should consider the
geographic features.The third standard is introduced to describe this
situation. Finally, it can be proved that the difference between the supporting
ratio of a certain party in each district and the average supporting ratio of
that particular party in the whole state obeys the Chi-square distribution
approximately. Consequently, we can obtain an archetypal formula to check
whether the gerrymandering we propose is fair.Comment: 23 pages,15 figures, 2007 American mathematical modeling contest
"Meritorious Winner
Periodic Lie Modules
Let be a prime number and be a positive integer not divisible by .
We describe the Heller translates of the periodic Lie module
in characteristic and show that it has period when is odd and
when
Sign sequences and decomposition numbers
We obtain a closed formula for the -decomposition numbers
arising from the canonical basis of the Fock space
representation of , where the partition
is obtained from by moving some nodes in its Young diagram, all of which
having the same -residue. We also show that when these -decomposition
numbers are evaluated at , we obtain the corresponding decomposition
numbers for the Schur algebras and symmetric groups
The Schur functor on tensor powers
Let be a left module for the Schur algebra , and let . Then is a -bimodule,
where the symmetric group on letters acts on the right by
place permutations. We show that the Schur functor sends to the -bimodule . As a corollary, we obtain the effect of the Schur functor on
the Lie power , symmetric power and exterior power
of .Comment: 6 page
The COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inflated count data
In this paper, we focus on the COM-type negative binomial distribution with
three parameters, which belongs to COM-type class distributions and
family of equilibrium distributions of arbitrary birth-death process. Besides,
we show abundant distributional properties such as overdispersion and
underdispersion, log-concavity, log-convexity (infinite divisibility), pseudo
compound Poisson, stochastic ordering and asymptotic approximation. Some
characterizations including sum of equicorrelated geometrically distributed
random variables, conditional distribution, limit distribution of COM-negative
hypergeometric distribution, and Stein's identity are given for theoretical
properties. COM-negative binomial distribution was applied to overdispersion
and ultrahigh zero-inflated data sets. With the aid of ratio regression, we
employ maximum likelihood method to estimate the parameters and the
goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.Comment: 22 pages,3 figures, Accepted for publication in Frontiers of
Mathematics in Chin
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