107 research outputs found
Generalized Foulkes modules and maximal and minimal constituents of plethysms of Schur functions
No full textThis paper proves a combinatorial rule giving all maximal and minimal partitions such that the Schur function appears in a plethysm of two arbitrary Schur functions. Determining the decomposition of these plethysms has been identified by Stanley as a key open problem in algebraic combinatorics. As corollaries we prove three conjectures of Agaoka on the partitions labeling the lexicographically greatest and least Schur functions appearing in an arbitrary plethysm. We also show that the multiplicity of the Schur function labelled by the lexicographically least constituent may be arbitrarily large. The proof is carried out in the symmetric group and gives an explicit non-zero homomorphism corresponding to each maximal or minimal partition
Character deflations and a generalization of the Murnaghan--Nakayama rule
No full textGiven natural numbers m and n, we define a deflation map from the characters of the symmetric group S_{mn} to the characters of S_n. This map is obtained by first restricting a character of S_{mn} to the wreath product S_m ?S_n, and then taking the sum of the irreducible constituents of the restricted character on which the base group S_m ×?×S_m acts trivially. We prove a combinatorial formula which gives the values of the images of the irreducible characters of S_{mn} under this map. We also prove an analogous result for more general deflation maps in which the base group is not required to act trivially. These results generalize the Murnaghan–Nakayama rule and special cases of the Littlewood–Richardson rule. As a corollary we obtain a new combinatorial formula for the character multiplicities that are the subject of the long-standing Foulkes' Conjecture. Using this formula we verify Foulkes' Conjecture in some new cases
On the structure of Foulkes modules for the symmetric group
No full textThis thesis concerns the structure of Foulkes modules for the symmetric group. We study `ordinary' Foulkes modules , where and are natural numbers, which are permutation modules arising from the action on cosets of . We also study a generalisation of these modules , labelled by a partition of , which we call generalised Foulkes modules.
Working over a field of characteristic zero, we investigate the module structure using semistandard homomorphisms. We identify several new relationships between irreducible constituents of and , where is a natural number, and also apply the theory to twisted Foulkes modules, which are labelled by , obtaining analogous results.
We make extensive use of character-theoretic techniques to study , the ordinary character afforded by the Foulkes module , and we draw conclusions about near-minimal constituents of in the case where is even. Further, we prove a recursive formula for computing character multiplicities of any generalised Foulkes character , and we decompose completely the character in the cases where has either two rows or two columns, or is a hook partition.
Finally, we examine the structure of twisted Foulkes modules in the modular setting. In particular, we answer questions about the structure of over fields of prime characteristic
Decompositions of some twisted Foulkes characters
No full textWe decompose the twisted Foulkes characters ϕ(2n)ν, or equivalently the plethysm sν∘s(2), in the cases where ν has either two rows or two columns, or is a hook partition
Minimal and maximal constituents of twisted Foulkes characters
No full textWe prove combinatorial rules that give the minimal and maximal partitions labelling the irreducible constituents of a family of characters for the symmetric group that generalize Foulkes permutation characters. Restated in the language of symmetric functions, our results determine all minimal and maximal partitions that label Schur functions appearing in the plethysms . As a corollary we prove
two conjectures of Agaoka on the lexicographically least constituents of the plethysms and
The cellular structure of wreath product algebras
No full textWe review the definitions and basic theory of cellular algebras as
developed in the papers of Graham and Lehrer and of K ?
onig and Xi.
We then introduce a reformulation of the concept of an iterated inflation
of cellular algebras (a concept due originally to K ?onig and Xi), which
we use to show that the Brauer algebra is cellular (following the work
of K ?onig and Xi). We then review the notion of the wreath product of
an algebra with a symmetric group, and apply our work on iterated
inflations to prove that the wreath product of a cellular algebra with a
symmetric group is in all cases cellular, and we obtain a description of
the cell modules of such a wreath product
Brauer algebras with parameter n = 2 acting on tensor space
No full textLet k be a field of prime characteristic p and E an n-dimensional vector space. We completely describe the tensor space E-r viewed as a module for the Brauer algebra B (k) (r,delta) with parameter delta=2 and n=2. This description shows that while the tensor space still affords Schur-Weyl duality, it typically is not filtered by cell modules, and thus will not be equal to a direct sum of Young modules as defined in Hartmann and Paget (Math Z 254:333-357, 2006). This is very different from the situation for group algebras of symmetric groups. Other results about the representation theory of these Brauer algebras are obtained, including a new description of a certain class of irreducible modules in the case when the characteristic is two
Geometry and topology of horofunction compactifications
No full textThis thesis concerns the global geometry and topology of the horofunction compactification of various metric spaces. In Chapters 4-6 we study the horofunction compactification of various homogeneous Finsler metric spaces, and establish a homeomorphism between the horofunction compactification and the dual unit ball in the tangent space at the base point. This homeomorphism establishes a one-to-one correspondence between the geometric parts of the horoboundary and the relative interiors of faces of the dual ball. In Chapter 7 we build on the work of Gutiérrez, and explore the topology and geometry of the horofunction compactification of infinite dimensional ℓᵖ spaces for 1 ≤ p < ∞. We show a clear disconnect between the global geometry and topology of the horofunction compactification in the infinite dimensional case versus the finite dimensional case. We also establish a marked difference in the behaviour of the horoboundary of ℓ¹ versus ℓᵖ for 1 < p < ∞. Chapter 8 deals with the horofunction compactification of infinite-dimensional spin factors considered as JB-algebras. We show that the exponential map extends to a geometry preserving homeomorphism on the boundary, mapping the horofunction compactification of the spin factor homeomorphically onto the horofunction compactification of the positive cone equipped with the Thompson metric. We conclude by showing that, considering an infinite dimensional Hilbert space as the tangent space at the identity of infinite dimensional real hyperbolic space, the exponential extends to a homeomorphism between the horofunction compactification of infinite dimensional Hilbert space and infinite dimensional real hyperbolic space
Some problems related to plethysm
No full textThis thesis is concerned with plethysm. It investigates certain plethysm coefficients and also studies a diagram algebra whose representation theory is related to plethysm. We use techniques involving plethystic semistandard Young tableaux in order to provide information about near-maximal constituents of the plethysm sν ◦sµ when µ = (m),(12 ) or (2, 1). We study further the case where µ = (12 ) by the means of a recursive formula of Law and Okitani. We study the ramified partition algebra, proving some new results about its representation theory. We show that the ramified partition algebra is a cellular algebra and investigate its cell modules. We show that the cell modules of the ramified partition algebra form a stratifying system, and hence prove an analogue of the Hemmer-Nakano theorem for this algebra. We give partial results on the semisimplicity of the ramified partition algebra over C, making a conjecture for the general case. Finally, we study the restriction of the cell modules for the ramified partition algebra to the partition algebra, investigating two filtrations and making progress on the decomposition of such modules
A family of modules with Specht and dual Specht filtrations
No full textWe study the permutation module arising from the action of the symmetric group S-2n, on the conjugacy class of fixed-point-free involutions, defined over an arbitrary field. The indecomposable direct summands of these modules are shown to possess filtrations by Specht modules and also filtrations by dual Specht modules. We see that these provide counterexamples to a conjecture by Hemmer. Twisted permutation modules are also considered, as is an application to the Brauer algebra
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