Littlewood Richardson coefficients for reflection groups
|
|
- Eustace Porter
- 6 years ago
- Views:
Transcription
1 Littlewood Richardson coefficients for reflection groups Arkady Berenstein and Edward Richmond* University of British Columbia Joint Mathematical Meetings Boston January 7, 2012 Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
2 1 Preliminaries on flag varieties 2 Algebraic approach to Schubert Calculus 3 Statement of results 4 Examples 5 The Nil-Hecke ring Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
3 Preliminaries on flag varieties Preliminaries: Schubert Calculus of G/B Let G be a Kac-Moody group over C (or a simple Lie group). Fix T B G a maximal torus and Borel subgroup of G. Let W := N(T )/T denote the Weyl group G. Let G/B be the flag variety (projective ind-variety). For any w W, we have the Schubert variety X w = BwB/B G/B. Denote the cohomology class of X w by σ w H 2l(w) (G/B). Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
4 Preliminaries on flag varieties Additively, we have that H (G/B) w W Z σ w. Goal (Schubert Calculus) Compute the structure (Littlewood-Richardson) coefficients c w u,v with respect to the Schubert basis defined by the product σ u σ v = w W c w u,vσ w. Note that if l(w) l(u) + l(v), then c w u,v = 0. For any w, u, v W, we have that c w u,v 0. (proofs are geometric) For example, if G is a finite Lie group, then the cardinality for a generic choice of g 1, g 2, g 3 G. g 1 X u g 2 X v g 3 X w0w = c w u,v Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
5 Algebraic approach to Schubert Calculus Algebraic approach to Schubert calculus Let A = A(G) denote the Cartan matrix of G. Alternatively, fix a finite index set I and let A = {a ij } be an I I matrix such that a ii = 2 a ij Z 0 if i j a ij = 0 a ji = 0. The matrix A defines an action of a Coxeter group W generated by reflections {s i } i I on the vector space V := Span C {α i } i I given by where α i, α j := a ij. s i (v) := v v, α i α i In particular, Coxeter groups of this type are crystallographic. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
6 Algebraic approach to Schubert Calculus Algebraic approach to Schubert calculus If we abandon the group G, we can consider a matrix A as follows: Fix a finite index set I and let A = {a ij } be an I I matrix such that a ii = 2 a ij R 0 if i j a ij = 0 a ji = 0. The matrix A defines an action of a Coxeter group W generated by reflections {s i } i I on the vector space V := Span C {α i } i I given by where α i, α j := a ij. s i (v) := v v, α i α i Every Coxeter group can be represented as above. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
7 Algebraic approach to Schubert Calculus Some examples G = SL(4) (type A 3 ) A = and W = S 4 (symmetric group) G = Sp(4) (type C 2 ) [ ] 2 2 A = and W = I (4) (dihedral group of 8 elements) G = ŜL(2) (affine type A) [ ] 2 2 A = and W = I ( ) (free dihedral group) Let ρ = 2 cos(π/5) [ ] 2 ρ A = and W = I ρ 2 2 (5) (dihedral group of 10 elements) Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
8 Algebraic approach to Schubert Calculus Notation on sequences and subsets Any sequence i := (i 1,..., i m ) I m has a corresponding element s i1 s im W. If s i1 s im is a reduced word of some w W, then we say that i R(w), the collection of reduced words. For each i I m and subset K = {k 1 < k 2 < < k n } of the interval [m] := {1, 2,..., m} let the subsequence i K := (i k1,..., i kn ) I n. We say a sequence i is admissible if i j i j+1 for all j [m 1]. Observe that any reduced sequence is admissible. (In general, the converse is false.) Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
9 Algebraic approach to Schubert Calculus Definition Let m > 0 and let K, L be subsets of [m] := {1, 2,..., m} such that K + L = m. We say that a bijection is bounded if φ(k) < k for each k K. φ : K [m] \ L Definition Given a reduced sequence i = (i 1,..., i m ) I m, we say that a bounded bijection φ : K [m] \ L is i-admissible if the sequence i L and the sequences i L(k) are admissible for all k K where L(k) := L φ(k k ). Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
10 Statement of results Recall that the Littlewood-Richardson coefficients c w u,v are defined by the product σ u σ v = w W c w u,v σ w. Theorem: Berenstein-R, 2010 Let u, v, w W such that l(w) = l(u) + l(v) and let i = (i 1,..., i m ) R(w). Then c w u,v = p φ where the summation is over all triples (û, ˆv, φ), where û, ˆv [m] such that iû R(u), iˆv R(v). φ : û ˆv [m] \ (û ˆv) is an i-admissible bounded bijection. The Theorem is still true without i-admissible. The Theorem generalizes to structure coefficients of T -equivariant cohomology H T (G/B). Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
11 Statement of results Definition For any k [m] and i = (i 1,..., i m ), denote α k := α ik bounded bijection φ : K [m] \ L we define the monomial p φ Z by the formula and s k := s ik. For any p φ := ( 1) K k K w k (α k ), α φ(k) where w k := r L(k) φ(k)<r<k where the product is taken in the natural order induced by the sequence [m] and if the product is empty, we set w k = 1. Also, if K =, then p φ = 1. Theorem: Berenstein-R, 2010 If the Cartan matrix A = (a ij ) satisfies a ij a ji 4 i j, then p φ 0 when φ is an i-admissible bounded bijection. s r Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
12 Examples Example: G = ŜL(2) Examples where G = ŜL(2), i = (1, 2, 1, 2) [ A = Compute c w u,v where w = s 1 s 2 s 1 s 2, u = v = s 1 s 2. Find û, ˆv [4] = {1, 2, 3, 4} For , we have with bounded bijections NOTE: φ 1 is not i-admissible û ˆv = {3, 4} and [4] \ (û ˆv) = {1, 2} φ 1 : (3, 4) (1, 2) φ 2 : (3, 4) (2, 1). ] Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
13 Examples Example: G = ŜL(2) Examples where G = ŜL(2), i = (1, 2, 1, 2) For , we have bounded bijections φ 1 : (3, 4) (1, 2) φ 2 : (3, 4) (2, 1). p φ1 = α 3, α 1 s 3 (α 4 ), α 2 p φ2 = α 3, α 2 s 2 s 3 (α 4 ), α 1 Totaling over all bounded bijections, we have c w u,v = α 3, α1 s 3 (α 4 ), α2 + α 3, α2 s 2 s 3 (α 4 ), α1 s 3 (α 4 ), α 2 s 3 (α 4 ), α = = 6 With only i-admissible terms. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
14 Examples Example: G = ŜL(2) Examples where G = ŜL(2), i = (1, 2, 1, 2, 1) Compute c w u,v where w = s 1 s 2 s 1 s 2 s 1, u = s 1 s 2 s 1, v = s 2 s 1. Find û, ˆv [5] = {1, 2, 3, 4, 5} Totaling over all bounded bijections, we have c w u,v = α 4, α2 s 4 (α 5 ), α3 + α 4, α3 s 3 s 4 (α 5 ), α2 + s 3 (α 4 ), α1 s 3 s 4 (α 5 ), α2 + s 3 (α 4 ), α2 s 2 s 3 s 4 (α 5 ), α1 s 2 (α 3 ), α1 s 4 (α 5 ), α 3 s 4 (α 5 ), α3 s 2 s 3 s 4 (α 5 ), α = = 10 With only i-admissible terms. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
15 Examples where G = Examples Example: G = ŜL(2) ŜL(2), i = (2, 1,..., 2, 1) }{{} 2n Question: Bounded bijections vs. i-admissible bounded bijections? c(n) := c w u,v, w = s 2 s 1 s 2 s }{{} 1, u = v = s 2 s }{{} 1 2n n #{Bounded bijections} #{i-admissible bounded bijections} c(2) 6 3 c(3) 20 7 c(4) c(5) c(6) c(n)?? largest coeff of (1 + x + x 2 ) n?? ( ) 2n Remark: c(n) = n Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
16 Examples Example: G is rank 2. General rank 2 group G. Let a, b 0, Define Chebyshev sequences [ A = 2 a b 2 ]. A k := ab k 1 A k 2 and B k := ba k 1 B k 2 where A 0 = B 0 = 0 and A 1 = B 1 = 1. Let u k = s 2 s 1 }{{} k Corollary: Binomial formula (Kitchloo, 2008) The rank 2 Littlewood-Richardson coefficients and v k = s 1 s }{{} 2. k A c un u k,u n k = c vn+1 n A 2 A 1 v k+1,u n k = (A k A 2 A 1 )(A n k A 2 A 1 ) B c vn v k,v n k = c un+1 n B 2 B 1 u k+1,v n k = (B k B 2 B 1 )(B n k B 2 B 1 ) Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
17 Examples Example: G = SL(4) Examples where G = SL(4), i = (3, 2, 1, 3, 2) Compute c w u,v where A = w = s 3 s 2 s 1 s 3 s 2, u = s 3 s 1 s 2 = s 1 s 3 s 2, v = s 3 s 1 = s 1 s 3. Totaling over all bounded bijections, we have c w u,v = α 3, α1 s 3 (α 4 ), α2 + α 3, α2 s 2 s 3 (α 4 ), α1 α 3, α2 α 3, α2 = = 1 In this case: {Bounded bijections}={i-admissible bounded bijections} In general, the i-admissible formula is not nonnegative. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
18 The Nil-Hecke ring Construction of Kostant and Kumar Recall the the matrix A gives an action of the Coxeter group W on V and thus W acts on the algebras S := S(V ) and Q := Q(V ) (polynomials and rational functions). Define with product structure Q W := Q C[W ] and a Q-linear coproduct (q 1 w 1 )(q 2 w 2 ) := q 1 w 1 (q 1 )w 1 w 2 : Q W Q W Q Q W by (qw) := qw w = w qw. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
19 The Nil-Hecke ring For any i I, define x i := 1 α i (s i 1). If i = (i 1,..., i m ) R(w), then define x w := x i1 x im. If A is a Cartan matrix of some Kac-Moody group G, then (by Kostant-Kumar 1986) x w is independent of i R(w) x 2 i = 0 i I. Define the Nil-Hecke ring H W := w W S x w Q W. We have that (H W ) H W S H W (Kostant-Kumar, 1986). Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
20 The Nil-Hecke ring Define the coproduct structure constants p w u,v S by (x w ) = p w u,v x u x v. u,v W Consider the T -equivariant cohomology ring HT (G/B) S σ w and L-R w W coefficients c w u,v S defined by the cup product σ u σ v = c w u,vσ w. w W Theorem: Kostant-Kumar 1986 The coefficients c w u,v = p w u,v. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, / 20
Research Statement. Edward Richmond. October 13, 2012
Research Statement Edward Richmond October 13, 2012 Introduction My mathematical interests include algebraic combinatorics, algebraic geometry and Lie theory. In particular, I study Schubert calculus,
More informationCOMBINATORIAL CURVE NEIGHBORHOODS FOR THE AFFINE FLAG MANIFOLD OF TYPE A 1 1
COMBINATORIAL CURVE NEIGHBORHOODS FOR THE AFFINE FLAG MANIFOLD OF TYPE A 1 1 LEONARDO C. MIHALCEA AND TREVOR NORTON Abstract. Let X be the affine flag manifold of Lie type A 1 1. Its moment graph encodes
More informationChow rings of Complex Algebraic Groups
Chow rings of Complex Algebraic Groups Shizuo Kaji joint with Masaki Nakagawa Workshop on Schubert calculus 2008 at Kansai Seminar House Mar. 20, 2008 Outline Introduction Our problem (algebraic geometory)
More informationPATTERN AVOIDANCE AND RATIONAL SMOOTHNESS OF SCHUBERT VARIETIES SARA C. BILLEY
PATTERN AVOIDANCE AND RATIONAL SMOOTHNESS OF SCHUBERT VARIETIES SARA C. BILLEY Let w be an element of the Weyl group S n, and let X w be the Schubert variety associated to w in the ag manifold SL n (C
More informationTHE LEIBNIZ FORMULA FOR DIVIDED DIFFERENCE OPERATORS ASSOCIATED TO KAC-MOODY ROOT SYSTEMS
THE LEIBNIZ FORMULA FOR DIVIDED DIFFERENCE OPERATORS ASSOCIATED TO KAC-MOODY ROOT SYSTEMS BY MATTHEW JASON SAMUEL A dissertation submitted to the Graduate School New Brunswick Rutgers, The State University
More informationEigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups
Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups Shrawan Kumar Talk given at AMS Sectional meeting held at Davidson College, March 2007 1 Hermitian eigenvalue
More informationParabolic subgroups Montreal-Toronto 2018
Parabolic subgroups Montreal-Toronto 2018 Alice Pozzi January 13, 2018 Alice Pozzi Parabolic subgroups Montreal-Toronto 2018 January 13, 2018 1 / 1 Overview Alice Pozzi Parabolic subgroups Montreal-Toronto
More informationEquivariant K -theory and hook formula for skew shape on d-complete set
Equivariant K -theory and hook formula for skew shape on d-complete set Hiroshi Naruse Graduate School of Education University of Yamanashi Algebraic and Enumerative Combinatorics in Okayama 2018/02/20
More information(Equivariant) Chern-Schwartz-MacPherson classes
(Equivariant) Chern-Schwartz-MacPherson classes Leonardo Mihalcea (joint with P. Aluffi) November 14, 2015 Leonardo Mihalcea (joint with P. Aluffi) () CSM classes November 14, 2015 1 / 16 Let X be a compact
More informationKAZHDAN LUSZTIG CELLS IN INFINITE COXETER GROUPS. 1. Introduction
KAZHDAN LUSZTIG CELLS IN INFINITE COXETER GROUPS MIKHAIL V. BELOLIPETSKY AND PAUL E. GUNNELLS 1. Introduction Groups defined by presentations of the form s 1,..., s n s 2 i = 1, (s i s j ) m i,j = 1 (i,
More informationON THE TOPOLOGY OF KAC-MOODY GROUPS CONTENTS
ON THE TOPOLOGY OF KAC-MOODY GROUPS NITU KITCHLOO ABSTRACT. We study the topology of spaces related to Kac-Moody groups. Given a split Kac-Moody group over C, let K denote the unitary form with maximal
More informationEQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE TEN: MORE ON FLAG VARIETIES
EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE TEN: MORE ON FLAG VARIETIES WILLIAM FULTON NOTES BY DAVE ANDERSON 1 A. Molev has just given a simple, efficient, and positive formula for the structure
More informationarxiv:math/ v3 [math.rt] 21 Jan 2004
CLUSTER ALGEBRAS III: UPPER BOUNDS AND DOUBLE BRUHAT CELLS arxiv:math/0305434v3 [math.rt] 21 Jan 2004 ARKADY BERENSTEIN, SERGEY FOMIN, AND ANDREI ZELEVINSKY Abstract. We develop a new approach to cluster
More informationTECHNICAL RESEARCH STATEMENT 1. INTRODUCTION
TECHNICAL RESEARCH STATEMENT EDWARD RICHMOND 1. INTRODUCTION My research program explores relationships between the combinatorics and geometry of flag varieties, Schubert varieties and Coxeter groups.
More informationComputers work exceptionally on Lie groups
Computers work exceptionally on Lie groups Shizuo Kaji Fukuoka University First Global COE seminar on Mathematical Research Using Computers at Kyoto University Oct. 24, 2008 S.Kaji (Fukuoka U.) Computers
More informationSARA BILLEY AND IZZET COSKUN
SINGULARITIES OF GENERALIZED RICHARDSON VARIETIES SARA BILLEY AND IZZET COSKUN Abstract. Richardson varieties play an important role in intersection theory and in the geometric interpretation of the Littlewood-Richardson
More informationCoxeter-Knuth Graphs and a signed Little Bijection
Coxeter-Knuth Graphs and a signed Little Bijection Sara Billey University of Washington http://www.math.washington.edu/ billey AMS-MAA Joint Meetings January 17, 2014 0-0 Outline Based on joint work with
More informationAFFINE WEYL GROUPS IN K-THEORY AND REPRESENTATION THEORY. Contents
AFFINE WEYL GROUPS IN K-THEORY AND REPRESENTATION THEORY CRISTIAN LENART AND ALEXANDER POSTNIKOV arxiv:math/0309207v3 [math.rt] 28 Jun 2005 Abstract. We give an explicit combinatorial Chevalley-type formula
More informationNOTES ON POINCARÉ SERIES OF FINITE AND AFFINE COXETER GROUPS
NOTES ON POINCARÉ SERIES OF FINITE AND AFFINE COXETER GROUPS VICTOR REINER Abstract. There are two famous formulae relating the Poincaré series of a finite/affine Weyl group to the degrees of fundamental
More informationarxiv:math/ v2 [math.qa] 12 Jun 2004
arxiv:math/0401137v2 [math.qa] 12 Jun 2004 DISCRETE MIURA OPERS AND SOLUTIONS OF THE BETHE ANSATZ EQUATIONS EVGENY MUKHIN,1 AND ALEXANDER VARCHENKO,2 Abstract. Solutions of the Bethe ansatz equations associated
More informationLECTURE 11: SOERGEL BIMODULES
LECTURE 11: SOERGEL BIMODULES IVAN LOSEV Introduction In this lecture we continue to study the category O 0 and explain some ideas towards the proof of the Kazhdan-Lusztig conjecture. We start by introducing
More informationBruhat order, rationally smooth Schubert varieties, and hyperplane arrangements
FPSAC 2010, San Francisco, USA DMTCS proc. AN, 2010, 833 840 Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements Suho Oh 1 and Hwanchul Yoo Department of Mathematics, Massachusetts
More informationToshiaki Shoji (Nagoya University) Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 1 / 1
Character sheaves on a symmetric space and Kostka polynomials Toshiaki Shoji Nagoya University July 27, 2012, Osaka Character sheaves on a symmetric space and Kostka polynomials July 27, 2012, Osaka 1
More informationA relative version of Kostant s theorem
A relative version of Kostant s theorem 1 University of Vienna Faculty of Mathematics Srni, January 2015 1 supported by project P27072 N25 of the Austrian Science Fund (FWF) This talk reports on joint
More informationTHE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES
Horiguchi, T. Osaka J. Math. 52 (2015), 1051 1062 THE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES TATSUYA HORIGUCHI (Received January 6, 2014, revised July 14, 2014) Abstract The main
More informationA COPRODUCT STRUCTURE ON THE FORMAL AFFINE DEMAZURE ALGEBRA BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG
A COPRODUCT STRUCTURE ON THE FORMAL AFFINE DEMAZURE ALGEBRA BAPTISTE CALMÈS, KIRILL ZAINOULLINE, AND CHANGLONG ZHONG Abstract. In the present paper, we generalize the coproduct structure on nil Hecke rings
More informationReflection Groups and Invariant Theory
Richard Kane Reflection Groups and Invariant Theory Springer Introduction 1 Reflection groups 5 1 Euclidean reflection groups 6 1-1 Reflections and reflection groups 6 1-2 Groups of symmetries in the plane
More informationAFFINE WEYL GROUPS IN K-THEORY AND REPRESENTATION THEORY. Contents
AFFINE WEYL GROUPS IN K-THEORY AND REPRESENTATION THEORY CRISTIAN LENART AND ALEXANDER POSTNIKOV Abstract. We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized
More informationCluster structures on open Richardson varieties and their quantizations
Cluster structures on open Richardson varieties and their quantizations Lie Theory and Its Applications in Physics XI, Varna, June 15-21, 2015 Milen Yakimov (Louisiana State Univ.) Joint work with Tom
More informationL(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that
ALGEBRAIC GROUPS 61 5. Root systems and semisimple Lie algebras 5.1. Characteristic 0 theory. Assume in this subsection that chark = 0. Let me recall a couple of definitions made earlier: G is called reductive
More informationGeometry of Schubert varieties and Demazure character formula
Geometry of Schubert varieties and Demazure character formula lectures by Shrawan Kumar during April, 2011 Hausdorff Research Institute for Mathematics Bonn, Germany notes written by Brandyn Lee 1 Notation
More informationThe tangent space to an enumerative problem
The tangent space to an enumerative problem Prakash Belkale Department of Mathematics University of North Carolina at Chapel Hill North Carolina, USA belkale@email.unc.edu ICM, Hyderabad 2010. Enumerative
More informationGeometric realization of PRV components and the Littlewood Richardson cone
Contemporary Mathematics Volume 490, 2009 Geometric realization of PRV components and the Littlewood Richardson cone Ivan Dimitrov and Mike Roth To Raja on the occasion of his 70 th birthday Abstract.
More informationEXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS
EXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS TAKESHI IKEDA AND HIROSHI NARUSE Abstract. We describe the torus-equivariant cohomology ring of isotropic Grassmannians by using a localization
More informationSmoothness of Schubert varieties via patterns in root subsystems
Advances in Applied Mathematics 34 (2005) 447 466 www.elsevier.com/locate/yaama Smoothness of Schubert varieties via patterns in root subsystems Sara Billey a,1,, Alexander Postnikov b,2 a Department of
More informationCombinatorics of the Cell Decomposition of Affine Springer Fibers
Combinatorics of the Cell Decomposition of Affine Springer Fibers Michael Lugo Virginia Tech Under the Advising of Mark Shimozono 28 April 2015 Contents 1 Affine Springer Fiber 2 Affine Symmetric Group
More informationReducibility of generic unipotent standard modules
Journal of Lie Theory Volume?? (??)???? c?? Heldermann Verlag 1 Version of March 10, 011 Reducibility of generic unipotent standard modules Dan Barbasch and Dan Ciubotaru Abstract. Using Lusztig s geometric
More informationRoot systems. S. Viswanath
Root systems S. Viswanath 1. (05/07/011) 1.1. Root systems. Let V be a finite dimensional R-vector space. A reflection is a linear map s α,h on V satisfying s α,h (x) = x for all x H and s α,h (α) = α,
More informationWeyl orbits of π-systems in Kac-Moody algebras
Weyl orbits of π-systems in Kac-Moody algebras Krishanu Roy The Institute of Mathematical Sciences, Chennai, India (Joint with Lisa Carbone, K N Raghavan, Biswajit Ransingh and Sankaran Viswanath) June
More informationMultiplicity free actions of simple algebraic groups
Multiplicity free actions of simple algebraic groups D. Testerman (with M. Liebeck and G. Seitz) EPF Lausanne Edinburgh, April 2016 D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity
More informationDefining equations for some nilpotent varieties
1 Defining equations for some nilpotent varieties Eric Sommers (UMass Amherst) Ben Johnson (Oklahoma State) The Mathematical Legacy of Bertram Kostant MIT June 1, 2018 Kostant s interest in the Buckyball
More informationSymplectic varieties and Poisson deformations
Symplectic varieties and Poisson deformations Yoshinori Namikawa A symplectic variety X is a normal algebraic variety (defined over C) which admits an everywhere non-degenerate d-closed 2-form ω on the
More informationH(G(Q p )//G(Z p )) = C c (SL n (Z p )\ SL n (Q p )/ SL n (Z p )).
92 19. Perverse sheaves on the affine Grassmannian 19.1. Spherical Hecke algebra. The Hecke algebra H(G(Q p )//G(Z p )) resp. H(G(F q ((T ))//G(F q [[T ]])) etc. of locally constant compactly supported
More informationPART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS
PART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS Contents 1. Regular elements in semisimple Lie algebras 1 2. The flag variety and the Bruhat decomposition 3 3. The Grothendieck-Springer resolution 6 4. The
More informationTITS SYSTEMS, PARABOLIC SUBGROUPS, PARABOLIC SUBALGEBRAS. Alfred G. Noël Department of Mathematics Northeastern University Boston, MA
TITS SYSTEMS, PARABOLIC SUBGROUPS, PARABOLIC SUBALGEBRAS Alfred G. Noël Department of Mathematics Northeastern University Boston, MA In this paper we give a brief survey of the basic results on B-N pairs
More informationADMISSIBLE W -GRAPHS. John R. Stembridge
ADMISSIBLE W -GRAPHS John R. Stemridge jrs@umich.edu Contents. General W -Graphs. Admissile W -Graphs 3. The Agenda 4. The Admissile Cells in Rank 5. Cominatorial Characterization 6 3 6 5 4 5 36 5 3 4
More informationCoxeter-Knuth Classes and a Signed Little Bijection
Coxeter-Knuth Classes and a Signed Little Bijection Sara Billey University of Washington Based on joint work with: Zachary Hamaker, Austin Roberts, and Benjamin Young. UC Berkeley, February, 04 Outline
More informationVanishing Ranges for the Cohomology of Finite Groups of Lie Type
for the Cohomology of Finite Groups of Lie Type University of Wisconsin-Stout, University of Georgia, University of South Alabama The Problem k - algebraically closed field of characteristic p > 0 G -
More informationEKT of Some Wonderful Compactifications
EKT of Some Wonderful Compactifications and recent results on Complete Quadrics. (Based on joint works with Soumya Banerjee and Michael Joyce) Mahir Bilen Can April 16, 2016 Mahir Bilen Can EKT of Some
More informationCombinatorial models for the variety of complete quadrics
Combinatorial models for the variety of complete quadrics Soumya D. Banerjee, Mahir Bilen Can, Michael Joyce October 21, 2016 Abstract We develop several combinatorial models that are useful in the study
More informationPERMUTATION REPRESENTATIONS ON SCHUBERT VARIETIES arxiv:math/ v2 [math.rt] 4 Jun 2007
PERMUTATION REPRESENTATIONS ON SCHUBERT VARIETIES arxiv:math/64578v2 [mathrt] 4 Jun 27 JULIANNA S TYMOCZKO Abstract This paper defines and studies permutation representations on the equivariant cohomology
More informationGoresky MacPherson Calculus for the Affine Flag Varieties
Canad. J. Math. Vol. 62 (2), 2010 pp. 473 480 doi:10.4153/cjm-2010-029-x c Canadian Mathematical Society 2010 Goresky MacPherson Calculus for the Affine Flag Varieties Zhiwei Yun Abstract. We use the fixed
More informationWhat elliptic cohomology might have to do with other generalized Schubert calculi
What elliptic cohomology might have to do with other generalized Schubert calculi Gufang Zhao University of Massachusetts Amherst Equivariant generalized Schubert calculus and its applications Apr. 28,
More informationON SOME PARTITIONS OF A FLAG MANIFOLD. G. Lusztig
ON SOME PARTITIONS OF A FLAG MANIFOLD G. Lusztig Introduction Let G be a connected reductive group over an algebraically closed field k of characteristic p 0. Let W be the Weyl group of G. Let W be the
More informationMath 249B. Geometric Bruhat decomposition
Math 249B. Geometric Bruhat decomposition 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ Φ, and let B be the unique
More informationEQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE ONE: PREVIEW
EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE ONE: PREVIEW WILLIAM FULTON NOTES BY DAVE ANDERSON 1 Let G be a Lie group acting on the left on a space X. Around 1960, Borel defined the equivariant
More informationFlag Manifolds and Representation Theory. Winter School on Homogeneous Spaces and Geometric Representation Theory
Flag Manifolds and Representation Theory Winter School on Homogeneous Spaces and Geometric Representation Theory Lecture I. Real Groups and Complex Flags 28 February 2012 Joseph A. Wolf University of California
More informationLittlewood Richardson polynomials
Littlewood Richardson polynomials Alexander Molev University of Sydney A diagram (or partition) is a sequence λ = (λ 1,..., λ n ) of integers λ i such that λ 1 λ n 0, depicted as an array of unit boxes.
More informationA SCHUBERT BASIS IN EQUIVARIANT ELLIPTIC COHOMOLOGY
A SCHUBERT BASIS IN EQUIVARIANT ELLIPTIC COHOMOLOGY CRISTIAN LENART AND KIRILL ZAINOULLINE Abstract. We address the problem of defining Schubert classes independently of a reduced word in equivariant elliptic
More informationCohomology theories on projective homogeneous varieties
Cohomology theories on projective homogeneous varieties Baptiste Calmès RAGE conference, Emory, May 2011 Goal: Schubert Calculus for all cohomology theories Schubert Calculus? Cohomology theory? (Very)
More informationSpherical varieties and arc spaces
Spherical varieties and arc spaces Victor Batyrev, ESI, Vienna 19, 20 January 2017 1 Lecture 1 This is a joint work with Anne Moreau. Let us begin with a few notations. We consider G a reductive connected
More informationTHE p-smooth LOCUS OF SCHUBERT VARIETIES. Let k be a ring and X be an n-dimensional variety over C equipped with the classical topology.
THE p-smooth LOCUS OF SCHUBERT VARIETIES GEORDIE WILLIAMSON ABSTRACT. These are notes from talks given at Jussieu (seminaire Chevalley), Newcastle and Aberdeen (ARTIN meeting). They are intended as a gentle
More informationZeta functions of buildings and Shimura varieties
Zeta functions of buildings and Shimura varieties Jerome William Hoffman January 6, 2008 0-0 Outline 1. Modular curves and graphs. 2. An example: X 0 (37). 3. Zeta functions for buildings? 4. Coxeter systems.
More information0 A. ... A j GL nj (F q ), 1 j r
CHAPTER 4 Representations of finite groups of Lie type Let F q be a finite field of order q and characteristic p. Let G be a finite group of Lie type, that is, G is the F q -rational points of a connected
More informationOn certain family of B-modules
On certain family of B-modules Piotr Pragacz (IM PAN, Warszawa) joint with Witold Kraśkiewicz with results of Masaki Watanabe Issai Schur s dissertation (Berlin, 1901): classification of irreducible polynomial
More informationGEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS
GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS ANA BALIBANU DISCUSSED WITH PROFESSOR VICTOR GINZBURG 1. Introduction The aim of this paper is to explore the geometry of a Lie algebra g through the action
More informationExtended groups and representation theory
Extended groups and representation theory Jeffrey Adams David Vogan University of Maryland Massachusetts Institute of Technology CUNY Representation Theory Seminar April 19, 2013 Outline Classification
More informationGeometry of Schubert Varieties RepNet Workshop
Geometry of Schubert Varieties RepNet Workshop Chris Spencer Ulrich Thiel University of Edinburgh University of Kaiserslautern 24 May 2010 Flag Varieties Throughout, let k be a fixed algebraically closed
More informationJoint work with Milen Yakimov. Kent Vashaw. November 19, Louisiana State University Prime Spectra of 2-Categories.
Joint work with Milen Yakimov Louisiana State University kvasha1@lsu.edu November 19, 2016 Overview 1 2 3 A 2-category is a category enriched over the category small categories. So a 2-category T has:
More informationarxiv: v1 [math.rt] 2 Oct 2012
On the Poincaré series of Kac-Moody Lie algebras arxiv:0.0648v [math.rt] Oct 0 Chunhua Jin, jinch@amss.ac.cn Academy of Mathematics and Systems Science, Chinese Academy of Sciences Beijing 0090, China
More informationAFFINE INSERTION AND PIERI RULES FOR THE AFFINE GRASSMANNIAN
AFFINE INSERTION AND PIERI RULES FOR THE AFFINE GRASSMANNIAN THOMAS LAM, LUC LAPOINTE, JENNIFER MORSE, AND MARK SHIMOZONO Abstract. We study combinatorial aspects of the Schubert calculus of the affine
More informationTHE ADJOINT REPRESENTATION IN RINGS OF FUNCTIONS
REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 1, Pages 182 189 (July 10, 1997) S 1088-4165(97)00029-0 THE ADJOINT REPRESENTATION IN RINGS OF FUNCTIONS ERIC SOMMERS
More informationON POINCARE POLYNOMIALS OF HYPERBOLIC LIE ALGEBRAS
ON POINCARE POLYNOMIALS OF HYPERBOLIC LIE ALGEBRAS Meltem Gungormez Dept. Physics, Fac. Science, Istanbul Tech. Univ. 34469, Maslak, Istanbul, Turkey e-mail: gungorm@itu.edu.tr arxiv:0706.2563v2 [math-ph]
More informationCatalan functions and k-schur positivity
Catalan functions and k-schur positivity Jonah Blasiak Drexel University joint work with Jennifer Morse, Anna Pun, and Dan Summers November 2018 Theorem (Haiman) Macdonald positivity conjecture The modified
More informationBILLEY-POSTNIKOV DECOMPOSITIONS AND THE FIBRE BUNDLE STRUCTURE OF SCHUBERT VARIETIES
BILLEY-POSTNIKOV DECOMPOSITIONS AND THE FIBRE BUNDLE STRUCTURE OF SCHUBERT VARIETIES EDWARD RICHMOND AND WILLIAM SLOFSTRA arxiv:1408.0084v2 [math.ag] 9 Feb 2017 Abstract. A theorem of Ryan and Wolper states
More informationMIURA OPERS AND CRITICAL POINTS OF MASTER FUNCTIONS arxiv:math/ v2 [math.qa] 12 Oct 2004
MIURA OPERS AND CRITICAL POINTS OF MASTER FUNCTIONS arxiv:math/03406v [math.qa] Oct 004 EVGENY MUKHIN, AND ALEXANDER VARCHENKO, Abstract. Critical points of a master function associated to a simple Lie
More informationA Study on Kac-Moody Superalgebras
ICGTMP, 2012 Chern Institute of Mathematics, Tianjin, China Aug 2o-26, 2012 The importance of being Lie Discrete groups describe discrete symmetries. Continues symmetries are described by so called Lie
More informationBackground on Chevalley Groups Constructed from a Root System
Background on Chevalley Groups Constructed from a Root System Paul Tokorcheck Department of Mathematics University of California, Santa Cruz 10 October 2011 Abstract In 1955, Claude Chevalley described
More informationThe Waring rank of the Vandermonde determinant
The Waring rank of the Vandermonde determinant Alexander Woo (U. Idaho) joint work with Zach Teitler(Boise State) SIAM Conference on Applied Algebraic Geometry, August 3, 2014 Waring rank Given a polynomial
More informationarxiv: v4 [math.rt] 9 Jun 2017
ON TANGENT CONES OF SCHUBERT VARIETIES D FUCHS, A KIRILLOV, S MORIER-GENOUD, V OVSIENKO arxiv:1667846v4 [mathrt] 9 Jun 217 Abstract We consider tangent cones of Schubert varieties in the complete flag
More informationSpecialized Macdonald polynomials, quantum K-theory, and Kirillov-Reshetikhin crystals
Specialized Macdonald polynomials, quantum K-theory, and Kirillov-Reshetikhin crystals Cristian Lenart Max-Planck-Institut für Mathematik, Bonn State University of New York at Albany Geometry Seminar,
More informationarxiv: v2 [math.ag] 14 Mar 2019
ALGEBRAIC AND GEOMETRIC PROPERTIES OF FLAG BOTT SAMELSON VARIETIES AND APPLICATIONS TO REPRESENTATIONS arxiv:1805.01664v2 [math.ag] 14 Mar 2019 NAOKI FUJITA, EUNJEONG LEE, AND DONG YOUP SUH Abstract. We
More informationResearch Statement. Nicholas Teff:
Research Statement Nicholas Teff: () The symmetric group S n is everywhere in mathematics. A fundamental example of this is the representation theory of S n. A representation is simply a vector space invariant
More informationCHAINS IN THE BRUHAT ORDER
CHAINS IN THE BRUHAT ORDER ALEXANDER POSTNIKOV AND RICHARD P. STANLEY arxiv:math.co/0502363 v1 16 Feb 2005 Abstract. We study a family of polynomials whose values express degrees of Schubert varieties
More informationMath 249B. Tits systems
Math 249B. Tits systems 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ + Φ, and let B be the unique Borel k-subgroup
More informationQuotients of Poincaré Polynomials Evaluated at 1
Journal of Algebraic Combinatorics 13 (2001), 29 40 c 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. Quotients of Poincaré Polynomials Evaluated at 1 OLIVER D. ENG Epic Systems Corporation,
More informationStaircase diagrams and the enumeration of smooth Schubert varieties
FPSAC 06 Vancouver, Canada DMTCS proc. BC, 06, 07 08 Staircase diagrams and the enumeration of smooth Schubert varieties Edward Richmond and William Slofstra Department of Mathematics, Oklahoma State University,
More informationCOMBINATORIAL MODELS FOR THE VARIETY OF COMPLETE QUADRICS
COMBINATORIAL MODELS FOR THE VARIETY OF COMPLETE QUADRICS SOUMYA D. BANERJEE, MAHIR BILEN CAN, AND MICHAEL JOYCE Abstract. We develop several combinatorial models that are useful in the study of the SL
More informationLOWER BOUNDARY HYPERPLANES OF THE CANONICAL LEFT CELLS IN THE AFFINE WEYL GROUP W a (Ãn 1) Jian-yi Shi
LOWER BOUNDARY HYPERPLANES OF THE CANONICAL LEFT CELLS IN THE AFFINE WEYL GROUP W a (Ãn 1) Jian-yi Shi Department of Mathematics, East China Normal University, Shanghai, 200062, China and Center for Combinatorics,
More informationAbout polynomiality of the Poisson semicentre for parabolic subalgebras
About polynomiality of the Poisson semicentre for parabolic subalgebras University of Saint-Etienne, ICJ, LYON - France The Canicular Days - Haifa - July 2017 - Celebrating A. Joseph s 75th birthday Aim.
More informationThe Lusztig-Vogan Bijection in the Case of the Trivial Representation
The Lusztig-Vogan Bijection in the Case of the Trivial Representation Alan Peng under the direction of Guangyi Yue Department of Mathematics Massachusetts Institute of Technology Research Science Institute
More informationTotal positivity in Schubert varieties
Comment. Math. Helv. 72 (1997) 128 166 0010-2571/97/010128-39 $ 1.50+0.20/0 c 1997 Birkhäuser Verlag, Basel Commentarii Mathematici Helvetici Total positivity in Schubert varieties Arkady Berenstein and
More informationA CHEVALLEY FORMULA IN THE QUANTUM K THEORY RING FOR THE G 2 FLAG MANIFOLD
A CHEVALLEY FORMULA IN THE QUANTUM K THEORY RING FOR THE G 2 FLAG MANIFOLD MARK E. LEWERS AND LEONARDO C. MIHALCEA Abstract. Let X be the (generalized) flag manifold for the Lie group of type G 2 and let
More informationSPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS
SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS DAN CIUBOTARU 1. Classical motivation: spherical functions 1.1. Spherical harmonics. Let S n 1 R n be the (n 1)-dimensional sphere, C (S n 1 ) the
More informationCover Page. Author: Yan, Qijun Title: Adapted deformations and the Ekedahl-Oort stratifications of Shimura varieties Date:
Cover Page The handle http://hdl.handle.net/1887/56255 holds various files of this Leiden University dissertation Author: Yan, Qijun Title: Adapted deformations and the Ekedahl-Oort stratifications of
More informatione j = Ad(f i ) 1 2a ij/a ii
A characterization of generalized Kac-Moody algebras. J. Algebra 174, 1073-1079 (1995). Richard E. Borcherds, D.P.M.M.S., 16 Mill Lane, Cambridge CB2 1SB, England. Generalized Kac-Moody algebras can be
More informationLECTURE 20: KAC-MOODY ALGEBRA ACTIONS ON CATEGORIES, II
LECTURE 20: KAC-MOODY ALGEBRA ACTIONS ON CATEGORIES, II IVAN LOSEV 1. Introduction 1.1. Recap. In the previous lecture we have considered the category C F := n 0 FS n -mod. We have equipped it with two
More informationLecture Notes Introduction to Cluster Algebra
Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Update: May 16, 2017 5 Review of Root Systems In this section, let us have a brief introduction to root system and finite Lie type classification
More informationCharacteristic classes and Invariants of Spin Geometry
Characteristic classes and Invariants of Spin Geometry Haibao Duan Institue of Mathematics, CAS 2018 Workshop on Algebraic and Geometric Topology, Southwest Jiaotong University July 29, 2018 Haibao Duan
More informationQuantum cohomology of homogeneous varieties: a survey Harry Tamvakis
Quantum cohomology of homogeneous varieties: a survey Harry Tamvakis Let G be a semisimple complex algebraic group and P a parabolic subgroup of G The homogeneous space X = G/P is a projective complex
More information