scholarly journals A Hessenberg Generalization of the Garsia-Procesi Basis for the Cohomology Ring of Springer Varieties

10.37236/425 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Aba Mbirika
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Group Action ◽  
Cohomology Ring ◽  
Geometric Construction ◽  
Nilpotent Operator ◽  
Hessenberg Varieties ◽  
New Evidence ◽  
Springer Variety ◽  
Two Parameter

The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976, T.A. Springer observed that this variety's cohomology ring carries a symmetric group action, and he offered a deep geometric construction of this action. Sixteen years later, Garsia and Procesi made Springer's work more transparent and accessible by presenting the cohomology ring as a graded quotient of a polynomial ring. They combinatorially describe an explicit basis for this quotient. The goal of this paper is to generalize their work. Our main result deepens their analysis of Springer varieties and extends it to a family of varieties called Hessenberg varieties, a two-parameter generalization of Springer varieties. Little is known about their cohomology. For the class of regular nilpotent Hessenberg varieties, we conjecture a quotient presentation for the cohomology ring and exhibit an explicit basis. Tantalizing new evidence supports our conjecture for a subclass of regular nilpotent varieties called Peterson varieties.

scholarly journals Poset Pinball, the Dimension Pair Algorithm, and Type A Regular Nilpotent Hessenberg Varieties

ISRN Geometry ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-34 ◽  
Author(s):  
Darius Bayegan ◽  
Megumi Harada
Keyword(s):  
Equivariant Cohomology ◽  
Cohomology Ring ◽  
Type A ◽  
Successful Outcome ◽  
Combinatorial Game ◽  
Schubert Polynomials ◽  
Hessenberg Varieties ◽  
Hessenberg Variety ◽  
Springer Variety ◽  
Special Case

We develop the theory of poset pinball, a combinatorial game introduced by Harada-Tymoczko to study the equivariant cohomology ring of a GKM-compatible subspace X of a GKM space; Harada and Tymoczko also prove that, in certain circumstances, a successful outcome of Betti poset pinball yields a module basis for the equivariant cohomology ring of X. First we define the dimension pair algorithm, which yields a successful outcome of Betti poset pinball for any type A regular nilpotent Hessenberg and any type A nilpotent Springer variety, considered as GKM-compatible subspaces of the flag variety. The algorithm is motivated by a correspondence between Hessenberg affine cells and certain Schubert polynomials which we learned from Insko. Second, in a special case of regular nilpotent Hessenberg varieties, we prove that our pinball outcome is poset-upper-triangular, and hence the corresponding classes form a HS1*(pt)-module basis for the S1-equivariant cohomology ring of the Hessenberg variety.

scholarly journals HIGHER CONJUGATION COHOMOLOGY IN COMMUTATIVE HOPF ALGEBRAS

2001 ◽  
Vol 44 (1) ◽  
pp. 19-26 ◽  
Author(s):  
M. D. Crossley ◽  
Sarah Whitehouse
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Symmetric Group ◽  
Group Action ◽  
Hopf Algebras ◽  
Homotopy Theory ◽  
Cohomology Ring ◽  
Stable Homotopy ◽  
Stable Homotopy Theory ◽  
Primary 16W30

AbstractLet $A$ be a graded, commutative Hopf algebra. We study an action of the symmetric group $\sSi_n$ on the tensor product of $n-1$ copies of $A$; this action was introduced by the second author in 1 and is relevant to the study of commutativity conditions on ring spectra in stable homotopy theory 2.We show that for a certain class of Hopf algebras the cohomology ring $H^*(\sSi_n;A^{\otimes n-1})$ is independent of the coproduct provided $n$ and $(n-2)!$ are invertible in the ground ring. With the simplest coproduct structure, the group action becomes particularly tractable and we discuss the implications this has for computations.AMS 2000 Mathematics subject classification: Primary 16W30; 57T05; 20C30; 20J06; 55S25

scholarly journals Degrees of Regular Sequences With a Symmetric Group Action

2019 ◽  
Vol 71 (03) ◽  
pp. 557-578
Author(s):  
Federico Galetto ◽  
Anthony Vito Geramita ◽  
David Louis Wehlau
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Group Action ◽  
Homogeneous Polynomials ◽  
Regular Sequences

AbstractWe consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible isomorphism types for these ideals. Following up on that work, we now analyze the possible degrees of the elements in such regular sequences. For each case of our classification, we provide some criteria guaranteeing the existence of regular sequences in certain degrees.

scholarly journals Domino tableaux, Schützenberger involution, and the symmetric group action

2000 ◽  
Vol 225 (1-3) ◽  
pp. 15-24 ◽  
Author(s):  
Arkady Berenstein ◽  
Anatol N. Kirillov
Keyword(s):  
Symmetric Group ◽  
Group Action ◽  
Domino Tableaux

An A∞-structure on the Cohomology Ring of the Symmetric Group S p with Coefficients in 𝔽 p

2013 ◽  
Vol 17 (5) ◽  
pp. 1553-1585
Author(s):  
Stephan Schmid
Keyword(s):  
Symmetric Group ◽  
Cohomology Ring

scholarly journals The Cohomology Rings of Regular Nilpotent Hessenberg Varieties in Lie Type A

2017 ◽  
Vol 2019 (17) ◽  
pp. 5316-5388 ◽  
Author(s):  
Hiraku Abe ◽  
Megumi Harada ◽  
Tatsuya Horiguchi ◽  
Mikiya Masuda
Keyword(s):  
Commutative Algebra ◽  
Cohomology Ring ◽  
Regular Sequence ◽  
Cohomology Rings ◽  
Generators And Relations ◽  
Hessenberg Varieties ◽  
Special Cases ◽  
Hessenberg Variety ◽  
Fixed Positive Integer ◽  
Special Case

AbstractLet $n$ be a fixed positive integer and $h: \{1,2,\ldots,n\} \rightarrow \{1,2,\ldots,n\}$ a Hessenberg function. The main results of this paper are two-fold. First, we give a systematic method, depending in a simple manner on the Hessenberg function $h$, for producing an explicit presentation by generators and relations of the cohomology ring $H^\ast({\mathrm{Hess}}(\mathsf{N},h))$ with ${\mathbb Q}$ coefficients of the corresponding regular nilpotent Hessenberg variety ${\mathrm{Hess}}(\mathsf{N},h)$. Our result generalizes known results in special cases such as the Peterson variety and also allows us to answer a question posed by Mbirika and Tymoczko. Moreover, our list of generators in fact forms a regular sequence, allowing us to use techniques from commutative algebra in our arguments. Our second main result gives an isomorphism between the cohomology ring $H^*({\mathrm{Hess}}(\mathsf{N},h))$ of the regular nilpotent Hessenberg variety and the $\mathfrak{S}_n$-invariant subring $H^*({\mathrm{Hess}}(\mathsf{S},h))^{\mathfrak{S}_n}$ of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the $\mathfrak{S}_n$-action on $H^*({\mathrm{Hess}}(\mathsf{S},h))$ defined by Tymoczko). Our second main result implies that $\mathrm{dim}_{{\mathbb Q}} H^k({\mathrm{Hess}}(\mathsf{N},h)) = \mathrm{dim}_{{\mathbb Q}} H^k({\mathrm{Hess}}(\mathsf{S},h))^{\mathfrak{S}_n}$ for all $k$ and hence partially proves the Shareshian–Wachs conjecture in combinatorics, which is in turn related to the well-known Stanley–Stembridge conjecture. A proof of the full Shareshian–Wachs conjecture was recently given by Brosnan and Chow, and independently by Guay–Paquet, but in our special case, our methods yield a stronger result (i.e., an isomorphism of rings) by more elementary considerations. This article provides detailed proofs of results we recorded previously in a research announcement [2].

scholarly journals Irreducible Modular Representations of the Reflection GroupG(m,1,n)

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
José O. Araujo ◽  
Tim Bratten ◽  
Cesar L. Maiarú
Keyword(s):  
Symmetric Group ◽  
Weyl Group ◽  
Reflection Group ◽  
Geometric Construction ◽  
Weyl Groups ◽  
Modular Representations ◽  
Complex Reflection ◽  
Complex Reflection Group ◽  
One Year ◽  
Combinatorial Methods

In an article published in 1980, Farahat and Peel realized the irreducible modular representations of the symmetric group. One year later, Al-Aamily, Morris, and Peel constructed the irreducible modular representations for a Weyl group of typeBn. In both cases, combinatorial methods were used. Almost twenty years later, using a geometric construction based on the ideas of Macdonald, first Aguado and Araujo and then Araujo, Bigeón, and Gamondi also realized the irreducible modular representations for the Weyl groups of typesAnandBn. In this paper, we extend the geometric construction based on the ideas of Macdonald to realize the irreducible modular representations of the complex reflection group of typeG(m,1,n).

scholarly journals The Symmetric Group Action on Rank-selected Posets of Injective Words

Order ◽  
2016 ◽  
Vol 35 (1) ◽  
pp. 47-56 ◽  
Author(s):  
Christos A. Athanasiadis
Keyword(s):  
Symmetric Group ◽  
Group Action

scholarly journals The Mod Two Cohomology of the Moduli Space of Rank Two Stable Bundles on a Surface and Skew Schur Polynomials

2019 ◽  
Vol 71 (03) ◽  
pp. 683-715 ◽  
Author(s):  
Christopher W. Scaduto ◽  
Matthew Stoffregen
Keyword(s):  
Riemann Surface ◽  
Group Action ◽  
Moduli Space ◽  
Class Group ◽  
Cohomology Ring ◽  
Mapping Class ◽  
Schur Polynomials ◽  
Stable Bundles ◽  
Cohomology Rings ◽  
Cup Product

AbstractWe compute cup-product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping-class group action.

scholarly journals A Combinatorial Model for the Decomposition of Multivariate Polynomial Rings as $S_n$-Modules

10.37236/8935 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Rosa Orellana ◽  
Michael Zabrocki
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Polynomial Rings ◽  
Invariant Polynomials ◽  
Multivariate Polynomial ◽  
Generating Set ◽  
Combinatorial Model

We consider the symmetric group $S_n$-module of the polynomial ring with $m$ sets of $n$ commuting variables and $m'$ sets of $n$ anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition $\lambda$ (a partition of $n$) is the number of multiset tableaux of shape $\lambda$ satisfying certain column and row strict conditions.  We also present a finite generating set for the ring of $S_n$ invariant polynomials of this ring. 

Sign in / Sign up

Export Citation Format

Share Document