scholarly journals A Combinatorial Approach to Specht Module Cohomology

2012 ◽  
Vol 19 (spec01) ◽  
pp. 777-786 ◽  
Author(s):  
David J. Hemmer
Keyword(s):  
Symmetric Group ◽  
Explicit Description ◽  
Degree Zero ◽  
Combinatorial Approach ◽  
Specht Module ◽  
Combinatorial Description

For a Specht module Sλ for the symmetric group Σd, the cohomology H i(Σd,Sλ) is known only in degree i = 0. We give a combinatorial criterion equivalent to the nonvanishing of the degree i = 1 cohomology, valid in odd characteristic. Our condition generalizes James' solution in degree zero. We apply this combinatorial description to give some computations of Specht module cohomology, together with an explicit description of the corresponding modules. Finally, we suggest some general conjectures that might be particularly amenable to proof using this description.

scholarly journals $(\ell,0)$-Carter Partitions, their Crystal-Theoretic Behavior and Generating Function

10.37236/854 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Chris Berg ◽  
Monica Vazirani
Keyword(s):  
Symmetric Group ◽  
Generating Function ◽  
Hecke Algebra ◽  
Specht Module ◽  
Combinatorial Description ◽  
Basic Representation ◽  
Crystal Graph ◽  
Generating Series ◽  
The One ◽  
Regular Partitions

In this paper we give an alternate combinatorial description of the "$(\ell,0)$-Carter partitions". The representation-theoretic significance of these partitions is that they indicate the irreducibility of the corresponding specialized Specht module over the Hecke algebra of the symmetric group. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas, which is in terms of hook lengths. We use our result to find a generating series which counts such partitions, with respect to the statistic of a partition's first part. We then apply our description of these partitions to the crystal graph $B(\Lambda_0)$ of the basic representation of $\widehat{{sl}_{\ell}}$, whose nodes are labeled by $\ell$-regular partitions. Here we give a fairly simple crystal-theoretic rule which generates all $(\ell,0)$-Carter partitions in the graph $B(\Lambda_0)$.

scholarly journals Multiplicities of Higher Lie Characters

2003 ◽  
Vol 75 (1) ◽  
pp. 9-21 ◽  
Author(s):  
Manfred Schocker
Keyword(s):  
Symmetric Group ◽  
Associative Algebra ◽  
Free Associative Algebra ◽  
Combinatorial Description ◽  
Irreducible Components ◽  
Witt Basis ◽  
Special Case ◽  
Regular Character

AbstractThe higher Lie characters of the symmetric group Sn arise from the Poincaré-Birkhoff-Witt basis of the free associative algebra. They are indexed by the partitions of n and sum up to the regular character of Sn. A combinatorial description of the multiplicities of their irreducible components is given. As a special case the Kraśkiewicz-Weyman result on the multiplicities of the classical Lie character is obtained.

scholarly journals Cohomology Classes of Interval Positroid Varieties and a Conjecture of Liu

10.37236/6960 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Brendan Pawlowski
Keyword(s):  
Symmetric Group ◽  
Finite Subset ◽  
Upper Bound ◽  
Cohomology Class ◽  
Symmetric Functions ◽  
A Priori ◽  
Specht Module ◽  
Group A ◽  
Schubert Classes ◽  
Stanley Symmetric Functions

To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we exhibit the appropriate diagram variety as a component in a degeneration of one of Knutson's interval positroid varieties (up to Grassmann duality). A priori, the cohomology classes of these interval positroid varieties are represented by affine Stanley symmetric functions. We give a different formula for these classes as ordinary Stanley symmetric functions, one with the advantage of being Schur-positive and compatible with inclusions between Grassmannians.

A COMBINATORIAL APPROACH TO LINKING NUMBERS IN RATIONAL HOMOLOGY SPHERES

2000 ◽  
Vol 09 (05) ◽  
pp. 703-711
Author(s):  
CARL J. STITZ
Keyword(s):  
Combinatorial Approach ◽  
Rational Homology ◽  
Characteristic Curves ◽  
Combinatorial Description ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Linking Numbers ◽  
Heegaard Diagram ◽  
Rational Homology Spheres ◽  
Matrix Invariants

In this paper we find a method to compute the classical Seifert-Threlfall linking number for rational homology spheres without using 2-chains bounded by the curves in question. By using a Heegaard diagram for the manifold, we describe link isotopy combinatorially using the three traditional Reidemeister moves along with a fourth move which is essentially a Kirby move along the characteristic curves. This result is mathematical folklore which we set in print. We then use this combinatorial description of link isotopy to develop and prove the invariance of linking numbers. Once the linking numbers are in place, matrix invariants such as the Alexander polynomial can be computed.

Wreath products along the period-doubling route to chaos

1999 ◽  
Vol 19 (6) ◽  
pp. 1617-1636 ◽  
Author(s):  
J. D. H. SMITH
Keyword(s):  
Wreath Product ◽  
Period Doubling ◽  
Periodic Points ◽  
Wreath Products ◽  
Explicit Description ◽  
Quadratic Maps ◽  
Combinatorial Description ◽  
Product Construction ◽  
Renormalization Operator ◽  
Route To Chaos

The wreath-product construction is used to give a complete combinatorial description of the dynamics of period-doubling quadratic maps leading to the Feigenbaum map. An explicit description of the action on periodic points uses the Thue–Morse sequence. In particular, a wreath-product construction of this sequence is given. The combinatorial renormalization operator on the period-doubling family of maps is invertible.

On a conjecture of Carter concerning irreducible Specht modules

1978 ◽  
Vol 83 (1) ◽  
pp. 11-17 ◽  
Author(s):  
G. D. James
Keyword(s):  
Irreducible Representation ◽  
Symmetric Group ◽  
Characteristic Zero ◽  
Irreducible Representations ◽  
Specht Module ◽  
Specht Modules ◽  
Definition Of

We study the question: Which ordinary irreducible representations of the symmetric group remain irreducible modulo a prime p?Let Sλ be the Specht module corresponding to the partition λ of n. The definition of Sλ is ‘independent of the field we are working over’. When the field has characteristic zero, Sλ is irreducible, and gives the ordinary irreducible representation of corresponding to the partition λ. Thus we are interested in the problem of whether or not Sλ is irreducible over a field of characteristic p.

scholarly journals On C-Bases, Partition Pairs and Filtrations for Induced or Restricted Specht Modules

2019 ◽  
Vol 26 (01) ◽  
pp. 161-180
Author(s):  
Christos A. Pallikaros
Keyword(s):  
Symmetric Group ◽  
Hecke Algebra ◽  
Specht Module ◽  
Specht Modules

We obtain alternative explicit Specht filtrations for the induced and the restricted Specht modules in the Hecke algebra of the symmetric group (defined over the ring A = ℤ[q1/2, q−1/2], where q is an indeterminate) using C-bases for these modules. Moreover, we provide a link between a certain C-basis for the induced Specht module and the notion of pairs of partitions.

scholarly journals Mirković–Vilonen polytopes and Khovanov–Lauda–Rouquier algebras

2016 ◽  
Vol 152 (8) ◽  
pp. 1648-1696 ◽  
Author(s):  
Peter Tingley ◽  
Ben Webster
Keyword(s):  
Lie Algebras ◽  
Finite Type ◽  
General Type ◽  
The Other ◽  
Explicit Description ◽  
Affine Grassmannian ◽  
Combinatorial Description ◽  
Extra Information ◽  
Klr Algebras ◽  
Affine Type

We describe how Mirković–Vilonen (MV) polytopes arise naturally from the categorification of Lie algebras using Khovanov–Lauda–Rouquier (KLR) algebras. This gives an explicit description of the unique crystal isomorphism between simple representations of KLR algebras and MV polytopes. MV polytopes, as defined from the geometry of the affine Grassmannian, only make sense in finite type. Our construction on the other hand gives a map from the infinity crystal to polytopes for all symmetrizable Kac–Moody algebras. However, to make the map injective and have well-defined crystal operators on the image, we must in general decorate the polytopes with some extra information. We suggest that the resulting ‘KLR polytopes’ are the general-type analogues of MV polytopes. We give a combinatorial description of the resulting decorated polytopes in all affine cases, and show that this recovers the affine MV polytopes recently defined by Baumann, Kamnitzer, and the first author in symmetric affine types. We also briefly discuss the situation beyond affine type.

scholarly journals Dirac cohomology, the projective supermodules of the symmetric group and the Vogan morphism

2018 ◽  
Vol 70 (2) ◽  
pp. 535-563 ◽  
Author(s):  
Kieran Calvert
Keyword(s):  
Symmetric Group ◽  
Hecke Algebras ◽  
Explicit Description ◽  
Weyl Groups ◽  
Explicit Model ◽  
Dirac Theory ◽  
Dirac Cohomology ◽  
Graded Modules ◽  
Projective Representations ◽  
Spectrum Data

Abstract We derive an explicit description of the genuine projective representations of the symmetric group Sn using Dirac cohomology and the branching graph for the irreducible genuine projective representations of Sn. Ciubotaru and He [D. Ciubotaru and X. He, Green polynomials of Weyl groups, elliptic pairings, and the extended index. Adv. Math., 283:1–50, 2015], using the extended Dirac index, showed that the characters of the projective representations of Sn are related to the characters of elliptic-graded modules. We derive the branching graph using Dirac theory and combinatorics relating to the cohomology of Borel varieties ℬe of g and are able to use Dirac cohomology to construct an explicit model for the projective representations. We also describe Vogan’s morphism for Hecke algebras in type A using spectrum data of the Jucys–Murphy elements.

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