1,099 research outputs found
(s,t)-cores: a weighted version of Armstrong's conjecture
The study of core partitions has been very active in recent years, with the
study of -cores - partitions which are both - and -cores - playing
a prominent role. A conjecture of Armstrong, proved recently by Johnson, says
that the average size of an -core, when and are coprime positive
integers, is . Armstrong also conjectured that the
same formula gives the average size of a self-conjugate -core; this was
proved by Chen, Huang and Wang.
In the present paper, we develop the ideas from the author's paper [J.
Combin. Theory Ser. A 118 (2011) 1525-1539] studying actions of affine
symmetric groups on the set of -cores in order to give variants of
Armstrong's conjectures in which each -core is weighted by the
reciprocal of the order of its stabiliser under a certain group action.
Informally, this weighted average gives the expected size of the -core of a
random -core
2-chains: An interesting family of posets
We introduce a new family of finite posets which we call 2-chains. These
first arose in the study of 0-Hecke algebras, but they admit a variety of
different characterisations. We give these characterisations, prove that they
are equivalent and derive some numerical results concerning 2-chains
Crystals, regularisation and the Mullineux map
The Mullineux map is a combinatorial function on partitions which describes the effect of tensoring a simple module for the symmetric group in characteristic with the one-dimensional sign representation. It can also be interpreted as a signed isomorphism between crystal graphs for . We give a new combinatorial description of the Mullineux map by expressing this crystal isomorphism as a composition of isomorphisms between different crystals. These isomorphisms are defined in terms of new generalised regularisation maps introduced by Millan Berdasco. We then given two applications of our new realisation of the Mullineux map, by providing purely combinatorial proofs of a conjecture of Lyle relating the Mullineux map with regularisation, and a theorem of Paget describing the Mullineux map in RoCK blocks of symmetric groups
Defect 2 spin blocks of symmetric groups and canonical basis coefficients
This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect
2
, analogous to Richardsās formula for defect
2
blocks of symmetric groups.
By developing a suitable analogue of the combinatorics used by Richards, we find a formula for the corresponding ā
q
-decomposition numbersā, i.e. the canonical basis coefficients in the level-
1
q
-deformed Fock space of type
A
2
ā¢
n
(
2
)
; a special case of a conjecture of Leclerc and Thibon asserts that these coefficients yield the spin decomposition numbers in characteristic
2
ā¢
n
+
1
. Along the way, we prove some general results on
q
-decomposition numbers. This paper represents the first substantial progress on canonical bases in type
A
2
ā¢
n
(
2
)
Simultaneous core multipartitions
We initiate the study of simultaneous core multipartitions, generalising
simultaneous core partitions, which have been studied extensively in the recent
literature. Given a multipartition datum (s|c), which consists of a
non-negative integer s and an l-tuple c of integers, we introduce the notion of
an (s|c)-core multipartition. Given an arbitrary set of multicore data, we give
necessary and sufficient conditions for the corresponding set of simultaneous
core multipartitions to be finite. We then study the special case of
simultaneous core bipartitions, giving exact enumerative results in some
special subcases.Comment: In this version the conjectures at the end of Section 4 have been
extende
Dyck tilings and the homogeneous Garnir relations for graded Specht modules
Suppose and are integer partitions with
. Kenyon and Wilson have introduced the notion of a
cover-inclusive Dyck tiling of the skew Young diagram ,
which has applications in the study of double-dimer models. We examine these
tilings in more detail, giving various equivalent conditions and then proving a
recurrence which we use to show that the entries of the transition matrix
between two bases for a certain permutation module for the symmetric group are
given by counting cover-inclusive Dyck tilings. We go on to consider the
inverse of this matrix, showing that its entries are determined by what we call
cover-expansive Dyck tilings. The fact that these two matrices are mutual
inverses allows us to recover the main result of Kenyon and Wilson.
We then discuss the connections with recent results of Kim et al, who give, a
simple expression for the sum, over all , of the number of cover-inclusive
Dyck tilings of . Our results provide a new proof of this
result. Finally, we show how to use our results to obtain simpler expressions
for the homogeneous Garnir relations for the universal Specht modules
introduced by Kleshchev, Mathas and Ram for the cyclotomic quiver Hecke
algebras
Evaluating three frameworks for the value of information: adaptation to task characteristics and probabilistic structure
We identify, and provide an integration of, three frameworks for measuring the
informativeness of cues in a multiple-cue judgment task. Cues can be ranked by information
value according to expected information gain (Bayesian framework), cue-outcome correlation
(Correlational framework), or ecological validity (Ecological framework). In three
experiments, all frameworks significantly predicted information acquisition, with the
Correlational (then the Bayesian) framework being most successful. Additionally,
participants adapted successfully to task characteristics (cue cost, time pressure, and
information limitations) ā altering the gross amount of information acquired, but not
responding to more subtle features of the cuesā information value that would have been
beneficial. Rational analyses of our task environments indicate that participants' behavior can
be considered successful from a boundedly rational standpoint
- ā¦