Computers work exceptionally on Lie groups

Size: px
Start display at page:

Download "Computers work exceptionally on Lie groups"

Transcription

1 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 work exceptionally on Lie groups GCOE seminar at Kyoto 1 / 1

2 Outline Introduction Computation of topological invariants s of Lie groups Schubert calculus (geometry) Divided difference operator (combinatorics) Borel presentation (algebraic topology) Computer assisted part Open problems S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 2 / 1

3 Introduction About presentation Latex-Beamer This presentation slide is made with LaTeX-Beamer. It is an easy-to-use L A TEXpackage, free of charge It produces a PDF file, which is almost environment independent There are a lot of people who use it; you can ask, consult web pages, even request new features It has the capability of and hyperlinks Main Theorem this kind of gimmicks S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 3 / 1

4 Introduction About presentation Latex-Beamer Theorem A mathematician is an optimist Proof. I have discovered a truly remarkable proof which this margin is too small to contain. S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 4 / 1

5 Introduction Computer assisted mathematics Advantages and Disadvantages Advantages of using computers are: Theorem can be proven while you are sleeping Everyone can confirm the computation The development of computers and softwares might produce new theorems Disadvantages of using computers are: It requires non-essential, non-mathematical work to make a practical program (sometimes we may have to struggle with bugs in compilers, OS, CPU...) It looks less elegant than human proof consequently, results are often underestimated by those who don t use computers Where and in what form can we submit results? S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 5 / 1

6 Common process Computation of topological invariants General setting Problem Compute some invariant F (X) for a space X, where F is a functor from spaces to some algebras Look for a theory which enables a concrete calculation Compute easy cases by hand to get insight Translate the mathematical theory into computer algorism Search for parts (libraries) made by others Implement of the data structure that corresponds to mathematical objects (polynomial, DGA, Lie algebra, Hopf algebra,... ) Choose programming language, software, etc. In algebraic topology, we usually resort to symbolic computation rather than numerical one Run the program and pray! Optimize it from both mathematical and computer s point of view Confirm the result by different algorism, softwares, platforms,... Look at the result to find general theory behind it S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 6 / 1

7 Lie group basic Computation of topological invariants Lie group G: simple, simply-connected, compact Lie group G T : maximal torus (dim T = l is the rank of G) t g: their Lie algebras with an invariant inner product (, ) t Π = {α i } 1 i l : simple roots 2α j {ω i } 1 i l : fundamental weights ( ( (α, ω j,α j) i) = δ ij ) H (BT ; Z) = Z[w 1,..., w l ], where w i = 2 s i GL(t ): simple reflection corresponding to α i, s i Aut(Z[w 1,..., w l ]) ( s i (e) = e ( 2αi (α, e)α i,α i) i ) W = N(T )/T : Weyl group of G (= a finite group generated by {s i } 1 i l ) Classification: A n, B n, C n, D n, G 2, F 4, E 6, E 7, E 8 These objects are all concrete (symbolic). However, for example, w i s are possibly irrational vectors so we need to handle with care. (Stembridge s coxeter/weyl package in Maple is so convenient that this often encourage me to choose Maple) S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 7 / 1

8 Computation of topological invariants Lie group Computable invariants rational cohomology is the invariant ring H (BG; Q) = H (BT ; Q) W mod p cohomology H (BG; F p ) the invariant rings H (BT ; F p ) W and H (B(Z/p) n ; Z/p) Wp rational cohomology of flag variety is the coinvariant ring H (G/T ; Q) = H (BT ; Q)/(H + (BT ; Q) W ) stable homotopy group π S (G) free resolution of H (G; F p ) as A p -algebra Grothendieck s torsion index of G Gröbner basis of (H + (BT ; Z) W ) H (BT ; Z) H (ΩG), H (G/P ), K (ΩG), cat(g),... self-equivalence of generalized flag variety Aut(G/P ) Aut(H (G/P ; Q)), where P is a parabolic subgroup S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 8 / 1

9 An example of invariants: The Chow ring A (G) S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 9 / 1

10 Notation Notation G: simply connected simple complex Lie group (SL(n, C)) B: Borel subgroup of G (the subgroup of upper triangular matrices) G/B: a projective variety called the flag variety (the space of flags, 0 V 1 V 2 V n 1 V n = C n, dim C (V i ) = i). H (G/B; Z): ordinary integral cohomology of G/B A (G): Chow ring of G A (G) = L i 0 Ai (G) A i (G) is a group of the rational equivalence classes of algebraic cycles of codimension i. (an algebraic cycle is a linear sum of possibly singular subvarieties) intersection product A i (G) A j (G) A i+j (G) S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 10 / 1

11 Chow ring of G Problem Goal Determine A (G) for all simply connected simple complex Lie groups Classification Theorem tells that G is one of the following: SL n, Spin n, Sp n, G 2, F 4, E 6, E 7, E 8 Grothendieck considered the problem in the 1950 s (Grothendieck) A (G) = H (G/B; Z)/(H 2 (G/B; Z)) Consequently, A (G) Q = Q for all G and A (G) = Z for G = SL n, Sp n A (G) for G = Spin n, G 2, F 4 were determined by R.Marlin(1974) Remaining cases are when G = E 6, E 7, E 8. S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 11 / 1

12 Grothendieck s Theorem Theory Theorem (Grothendieck(1958)) the cycle map cl : A (G/B) H 2 (G/B; Z) is an isomorphism of rings: A (G/B) H 2 (G/B; Z) the pullback of the projection p : G G/B induces a surjection p : A (G/B) A (G), where the kernel is an ideal generated by A 1 (G/B). Corollary A (G) = H (G/B; Z)/(H 2 (G/B; Z)) A (G) = Z, G = SL n, Sp n S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 12 / 1

13 Schubert class Schubert calculus The Bruhat decomposition gives a cell decomposition G/B = BwB/B w W X w = closure of BwB/B( = C l(w) ): Schubert variety Z w = {the cohomology class corresponding to [X w0w]} H 2l(w) (G/B; Z): Schubert class {Z w } w W forms an additive basis for H (G/B; Z) (indexed by W ) In particular, H (G/B; Z) is torsion free S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 13 / 1

14 Schubert calculus Structure constant The intersection product of two Schubert classes Z w, Z w can be written in the linear sum of Schubert classes: Z w Z w = c v ww Z v The coefficients c v ww A goal in Schubert calculus Give a combinatorial formula for c v ww l(v)=l(w)+l(w ) Z are called the structure constants Littlewood-Richardson rule for Grassmaniann Chevalley formula for Z w Z w when l(w) = 1 Schubert polynomial S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 14 / 1

15 Divided difference operator Combinatorial machinery (translator) Definition (B-G-G(1973), Demazure(1973)) 1 For α i Π, i : H (BT ; Z) H 2 (BT ; Z) i (f) = f s i(f) α i, f H (BT ; Z) = Z[ω 1,..., ω l ] 2 For w W, w = s i1 s i2 s ik : a reduced decomposition, w = i1 i2 ik : H (BT ; Z) H 2k (BT ; Z) S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 15 / 1

16 Combinatorial machinery (translator) Theorem (B-G-G(1973), Demazure(1973)) the characteristic map c : H 2k (BT ; Z) H 2k (G/B; Z): c(f) = X w(f)z w (Note: w(f) Z) l(w)=k the following composition is induced by the inclusion of Z Q: H (G/B; Z) H (G/B; Q) = H (BT ; Q)/H + (BT ; Q) W c H (G/B; Q) (Giambelli formula) Q ««α α Z w = c + w 1w0 W Inductive formula: α(ω β ) = δ αβ α(fg) = α(f)g + s α(f) α(g) w = i1 i2 ik S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 16 / 1

17 Combinatorial machinery (translator) Ring structure of H (G/B; Z) polynomial characteristic map Giambelli formula 1 Given elements Z w, Z w H (G/B; Z) Schubert classes (Weyl group) 2 by Giambelli formula, we have polynomials f, f H (BT ; Q) which correspond to Z w, Z w 3 by characteristic map, we have c(f f ) H (G/B; Z) which correspond to the intersection product Z w Z w 4 we obtain the ring structure of H (G/B; Z) 5 hence we obtain A(G) = H (G/B; Z)/H 2 (G/B; Z) S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 17 / 1

18 Optimization Computational complexity characteristic map: Giambelli formula: c(f) = l(w)=k w (f)z w ( α Z w = c ( α )) + w 1w0 W For G = E 8, W = = l(w 0 ) = + = 1 2 dim G/B = 120 Today s computer cannot handle polynomials of degree 120! The above strategy is not practical as it is S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 18 / 1

19 Borel presentation Optimization There are two descriptions for H (G/B; Z) = A (G/B) Borel presentation Schubert presentation quotient of a polynomial ring Z-basis indexed by Weyl group elements polynomials Schubert classes geometry no algebraic cycles ring structure easy hard (main theme of Schubert calculus) Borel presentation characteristic map Schubert presentation Giambelli formula S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 19 / 1

20 Optimization Results from algebraic topology By spectral sequence argument, Borel presentation for each H (G/B; Z) was computed by Borel, Toda-Watanabe, Bott-Samelson, and Nakagawa. They have the following form in general. H (G/B; Z) = Z[t i, γ j ]/(ideal), ( t i = 2, γ j > 2 Our strategy is: 1 Compute A (G) purely algebraically from Borel presentation A (G) = H (G/B; Z)/(H 2 (G/B; Z)) = Z[γ j ]/(ideal) 2 Find Schubert varieties representing the generators γ j S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 20 / 1

21 More optimization for type E Convenient presentation of H (BT ; Z) Let G be either E 6, E 7, or E 8. α 1 α 3 α 4 α 5 α 6 α 2 If we take away α 2, then the Dynkin diagram becomes type A. By this observation, We take another set of generators for H (BT ; Z) = Z[ω 1, ω 2,..., ω l ]: t l = ω l t i = s i+1 (t i+1 ) = t 1 = s 1 (t 2 ) = ω 1 + ω 2 t = ω 2 { ω i ω i+1 (4 i l 1) ω i 1 + ω i ω i+1 (i = 2, 3) S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 21 / 1

22 More optimization for type E Toda-Watanabe s magical basis Let c i = i-th elementary symmetric function in t 1,..., t l (1 i l) H (BT ; Z) = Z[ω 1, ω 2,..., ω l ] = Z[t 1, t 2,..., t l, t]/(c 1 3t). s i (i 2) act on {t i } 1 i l as permutations and trivially on t. For example, for f Z[t, c 2,..., c l ], i f = 0 if i 2 this reduces the computation S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 22 / 1

23 Ingredients from algebraic topology Theorem (Nakagawa(2001)) H (E 7 /B; Z) = Z[t 1, t 2,..., t 7, t, γ 3, γ 4, γ 5, γ 9 ] /(ρ 1, ρ 2, ρ 3, ρ 4, ρ 5, ρ 6, ρ 8, ρ 9, ρ 10, ρ 12, ρ 14, ρ 18 ), ρ 1 = c 1 3t, ρ 2 = c 2 4t 2, ρ 3 = c 3 2γ 3, ρ 4 = c 4 + 2t 4 3γ 4, ρ 5 = c 5 3tγ 4 + 2t 2 γ 3 2γ 5, ρ 6 = γ c 6 2tγ 5 3t 2 γ 4 + t 6, ρ 8 = 3γ 4 2 2γ 3 γ 5 + t(2c 7 6γ 3 γ 4 ) 9t 2 c t 3 γ t 4 γ 4 6t 5 γ 3 t 8, S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 23 / 1

24 Ingredients from algebraic topology From the result in previous slide, an easy calculation by hand shows A (E 7 ) = A (E 7 /B)/(A 1 (E 7 /B)) = H (E 7 /B; Z)/(H 2 (E 7 /B; Z)) = Z[γ 3, γ 4, γ 5, γ 9 ]/(2γ 3, 3γ 4, 2γ 5, γ 2 3, 2γ 9, γ 2 5, γ 3 4, γ 2 9) By using a Maple script, we obtain γ 3 = Z Z 542 γ 4 = Z Z Z 6542 γ 5 = Z γ 9 = 2Z Z Note that we abbreviate s i1 s i2 s ik W as i 1 i k A (E 7 ) = Z[Z 542, Z 6542, Z 76542, Z ] / 2Z 542, 3Z 6542, 2Z 76542, Z542 2, 2Z , Z , Z3 6542, Z S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 24 / 1

25 Final results Theorem (K-Nakagawa) A(E 6 ) = Z[Z 542, Z 6542 ]/(2Z 542, 3Z 6542, Z 2 542, Z ), (Z 542 = B(w 0 s 6 s 5 s 4 s 2 )B G etc.) A(E 7 ) = Z[X 3, X 4, X 5, X 9 ] /(2X 3, 3X 4, 2X 5, X 2 3, 2X 9, X 2 5, X 3 4, X 2 9 ) (X 3 = Z 542, X 4 = Z 6542, X 5 = Z 76542, X 9 = Z ) A(E 8 ) = Z[X 3, X 4, X 5, X 6, X 9, X 10, X 15 ] / 2X 3, 3X 4, 2X 5, 5X 6, 2X 9, X5 2 3X 10, X4 3, 2X 15, X9 2, 3X2 10, X8 3, X X X5 6 S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 25 / 1

26 Computation to theory We encounter spin-off problems during the process. Definition Z w is indecomposable Z w Z v l(v)<l(w) torsion index and decomposability t(g) = min{t t Z w Im(c), w W } combinatorics of Weyl group and decomposability Schubert polynomial for exceptional types (Schubert polynomials live in H (BT ; Z) Z[γ i ], where γ i s correspond to indecomposable Schubert classes) Cohomology ( Chow rings ) of generalized flag varieties G/P I hope further experimentation and visualization will lead to the solution. S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 26 / 1

27 Open problems Open problems Cohomology ring Invariant ring of Weyl group Z[w1,..., w l ] W Invariant ring of mod p Weyl group Fp[v 1,..., v m] Wp Action of Steenrod operations on H (BG; F p) Homotopy groups Handy free resolutions for algebras which arose as cohomology rings Homology of the λ-algebra (E2-term of Adams spectral sequence) How to handle, for example, algebras over Steenrod algebras? or generally, algebras over operads? How to deal with spectral sequences? S.Kaji (Fukuoka U.) Computers work exceptionally on Lie groups GCOE seminar at Kyoto 27 / 1

Chow rings of Complex Algebraic Groups

Chow 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 information

LECTURE 11: SOERGEL BIMODULES

LECTURE 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 information

(Equivariant) Chern-Schwartz-MacPherson classes

(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 information

Cohomology theories on projective homogeneous varieties

Cohomology 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 information

A note on Samelson products in the exceptional Lie groups

A note on Samelson products in the exceptional Lie groups A note on Samelson products in the exceptional Lie groups Hiroaki Hamanaka and Akira Kono October 23, 2008 1 Introduction Samelson products have been studied extensively for the classical groups ([5],

More information

EKT of Some Wonderful Compactifications

EKT 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 information

Characteristic classes in the Chow ring

Characteristic classes in the Chow ring arxiv:alg-geom/9412008v1 10 Dec 1994 Characteristic classes in the Chow ring Dan Edidin and William Graham Department of Mathematics University of Chicago Chicago IL 60637 Let G be a reductive algebraic

More information

NOTES ON POINCARÉ SERIES OF FINITE AND AFFINE COXETER GROUPS

NOTES 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 information

Littlewood Richardson coefficients for reflection groups

Littlewood Richardson coefficients for reflection groups 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

More information

Parabolic subgroups Montreal-Toronto 2018

Parabolic 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 information

Background on Chevalley Groups Constructed from a Root System

Background 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 information

Chern Classes and the Chern Character

Chern Classes and the Chern Character Chern Classes and the Chern Character German Stefanich Chern Classes In this talk, all our topological spaces will be paracompact Hausdorff, and our vector bundles will be complex. Let Bun GLn(C) be the

More information

Equivariant 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 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

Kraśkiewicz-Pragacz modules and some positivity properties of Schubert polynomials

Kraśkiewicz-Pragacz modules and some positivity properties of Schubert polynomials Kraśkiewicz-Pragacz modules and some positivity properties of Schubert polynomials Masaki Watanabe Graduate School of Mathematical Sciences, the University of Tokyo December 9, 2015 Masaki Watanabe KP

More information

ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS

ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS LISA CARBONE Abstract. We outline the classification of K rank 1 groups over non archimedean local fields K up to strict isogeny,

More information

Another proof of the global F -regularity of Schubert varieties

Another proof of the global F -regularity of Schubert varieties Another proof of the global F -regularity of Schubert varieties Mitsuyasu Hashimoto Abstract Recently, Lauritzen, Raben-Pedersen and Thomsen proved that Schubert varieties are globally F -regular. We give

More information

Intersection of stable and unstable manifolds for invariant Morse functions

Intersection of stable and unstable manifolds for invariant Morse functions Intersection of stable and unstable manifolds for invariant Morse functions Hitoshi Yamanaka (Osaka City University) March 14, 2011 Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and

More information

A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds

A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds arxiv:math/0312251v1 [math.dg] 12 Dec 2003 A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds Haibao Duan Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, dhb@math.ac.cn

More information

EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE ONE: PREVIEW

EQUIVARIANT 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 information

Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups

Eigenvalue 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 information

For Which Pseudo Reflection Groups Are the p-adic Polynomial Invariants Again a Polynomial Algebra?

For Which Pseudo Reflection Groups Are the p-adic Polynomial Invariants Again a Polynomial Algebra? Journal of Algebra 214, 553 570 (1999) Article ID jabr.1998.7691, available online at http://www.idealibrary.com on For Which Pseudo Reflection Groups Are the p-adic Polynomial Invariants Again a Polynomial

More information

THE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES

THE 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 information

Highest-weight Theory: Verma Modules

Highest-weight Theory: Verma Modules Highest-weight Theory: Verma Modules Math G4344, Spring 2012 We will now turn to the problem of classifying and constructing all finitedimensional representations of a complex semi-simple Lie algebra (or,

More information

THE 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. 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 information

Symplectic varieties and Poisson deformations

Symplectic 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 information

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.

More information

Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements

Bruhat 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 information

Research Statement. Edward Richmond. October 13, 2012

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 information

Combinatorial models for the variety of complete quadrics

Combinatorial 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 information

Math 249B. Geometric Bruhat decomposition

Math 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 information

0 A. ... A j GL nj (F q ), 1 j r

0 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 information

Multiplicity free actions of simple algebraic groups

Multiplicity 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 information

16.2. Definition. Let N be the set of all nilpotent elements in g. Define N

16.2. Definition. Let N be the set of all nilpotent elements in g. Define N 74 16. Lecture 16: Springer Representations 16.1. The flag manifold. Let G = SL n (C). It acts transitively on the set F of complete flags 0 F 1 F n 1 C n and the stabilizer of the standard flag is the

More information

CALTECH ALGEBRAIC GEOMETRY SEMINAR: A GEOMETRIC LITTLEWOOD-RICHARDSON RULE RAVI VAKIL

CALTECH ALGEBRAIC GEOMETRY SEMINAR: A GEOMETRIC LITTLEWOOD-RICHARDSON RULE RAVI VAKIL CALTECH ALGEBRAIC GEOMETRY SEMINAR: A GEOMETRIC LITTLEWOOD-RICHARDSON RULE RAVI VAKIL ABSTRACT. I will describe an explicit geometric Littlewood-Richardson rule, interpreted as deforming the intersection

More information

Generically split projective homogeneous varieties

Generically split projective homogeneous varieties Generically split projective homogeneous varieties Viktor Petrov, Nikita Semenov Abstract Let G be an exceptional simple algebraic group over a field k and X a projective G-homogeneous variety such that

More information

EQUIVARIANT COHOMOLOGY. p : E B such that there exist a countable open covering {U i } i I of B and homeomorphisms

EQUIVARIANT COHOMOLOGY. p : E B such that there exist a countable open covering {U i } i I of B and homeomorphisms EQUIVARIANT COHOMOLOGY MARTINA LANINI AND TINA KANSTRUP 1. Quick intro Let G be a topological group (i.e. a group which is also a topological space and whose operations are continuous maps) and let X be

More information

COMBINATORIAL 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 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 information

L(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

L(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 information

The Real Grassmannian Gr(2, 4)

The Real Grassmannian Gr(2, 4) The Real Grassmannian Gr(2, 4) We discuss the topology of the real Grassmannian Gr(2, 4) of 2-planes in R 4 and its double cover Gr + (2, 4) by the Grassmannian of oriented 2-planes They are compact four-manifolds

More information

IVAN LOSEV. Lemma 1.1. (λ) O. Proof. (λ) is generated by a single vector, v λ. We have the weight decomposition (λ) =

IVAN LOSEV. Lemma 1.1. (λ) O. Proof. (λ) is generated by a single vector, v λ. We have the weight decomposition (λ) = LECTURE 7: CATEGORY O AND REPRESENTATIONS OF ALGEBRAIC GROUPS IVAN LOSEV Introduction We continue our study of the representation theory of a finite dimensional semisimple Lie algebra g by introducing

More information

SARA BILLEY AND IZZET COSKUN

SARA 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 information

Lecture on Equivariant Cohomology

Lecture on Equivariant Cohomology Lecture on Equivariant Cohomology Sébastien Racanière February 20, 2004 I wrote these notes for a hours lecture at Imperial College during January and February. Of course, I tried to track down and remove

More information

arxiv: v4 [math.rt] 9 Jun 2017

arxiv: 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 information

LECTURE 4.5: SOERGEL S THEOREM AND SOERGEL BIMODULES

LECTURE 4.5: SOERGEL S THEOREM AND SOERGEL BIMODULES LECTURE 4.5: SOERGEL S THEOREM AND SOERGEL BIMODULES DMYTRO MATVIEIEVSKYI Abstract. These are notes for a talk given at the MIT-Northeastern Graduate Student Seminar on category O and Soergel bimodules,

More information

arxiv: v1 [math.ag] 17 Apr 2015

arxiv: v1 [math.ag] 17 Apr 2015 FREE RESOLUTIONS OF SOME SCHUBERT SINGULARITIES. MANOJ KUMMINI, V. LAKSHMIBAI, PRAMATHANATH SASTRY, AND C. S. SESHADRI arxiv:1504.04415v1 [math.ag] 17 Apr 2015 Abstract. In this paper we construct free

More information

Math 249B. Tits systems

Math 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 information

STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY

STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY ALEXANDER MERKURJEV 1. Introduction Let p be a prime integer. For a pair of topological spaces A X we write H i (X, A; Z/pZ) for the i-th singular cohomology group

More information

Quantum cohomology of homogeneous varieties: a survey Harry Tamvakis

Quantum 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

Notation. For any Lie group G, we set G 0 to be the connected component of the identity.

Notation. For any Lie group G, we set G 0 to be the connected component of the identity. Notation. For any Lie group G, we set G 0 to be the connected component of the identity. Problem 1 Prove that GL(n, R) is homotopic to O(n, R). (Hint: Gram-Schmidt Orthogonalization.) Here is a sequence

More information

Combinatorics for algebraic geometers

Combinatorics for algebraic geometers Combinatorics for algebraic geometers Calculations in enumerative geometry Maria Monks March 17, 214 Motivation Enumerative geometry In the late 18 s, Hermann Schubert investigated problems in what is

More information

GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS

GEOMETRIC 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 information

Geometry of Schubert Varieties RepNet Workshop

Geometry 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 information

Representation theory and homological stability

Representation theory and homological stability Representation theory and homological stability Thomas Church and Benson Farb October 5, 2011 Abstract We introduce the idea of representation stability (and several variations) for a sequence of representations

More information

Parameterizing orbits in flag varieties

Parameterizing orbits in flag varieties Parameterizing orbits in flag varieties W. Ethan Duckworth April 2008 Abstract In this document we parameterize the orbits of certain groups acting on partial flag varieties with finitely many orbits.

More information

Characteristic classes and Invariants of Spin Geometry

Characteristic 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 information

ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS

ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need

More information

Math 249B. Nilpotence of connected solvable groups

Math 249B. Nilpotence of connected solvable groups Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C

More information

Math 210C. The representation ring

Math 210C. The representation ring Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let

More information

Cell decompositions and flag manifolds

Cell decompositions and flag manifolds Cell decompositions and flag manifolds Jens Hemelaer October 30, 2014 The main goal of these notes is to give a description of the cohomology of a Grassmannian (as an abelian group), and generalise this

More information

1.1 Definition of group cohomology

1.1 Definition of group cohomology 1 Group Cohomology This chapter gives the topological and algebraic definitions of group cohomology. We also define equivariant cohomology. Although we give the basic definitions, a beginner may have to

More information

Transcendental L 2 -Betti numbers Atiyah s question

Transcendental L 2 -Betti numbers Atiyah s question Transcendental L 2 -Betti numbers Atiyah s question Thomas Schick Göttingen OA Chennai 2010 Thomas Schick (Göttingen) Transcendental L 2 -Betti numbers Atiyah s question OA Chennai 2010 1 / 24 Analytic

More information

BUNDLES, COHOMOLOGY AND TRUNCATED SYMMETRIC POLYNOMIALS

BUNDLES, COHOMOLOGY AND TRUNCATED SYMMETRIC POLYNOMIALS BUNDLES, COHOMOLOGY AND TRUNCATED SYMMETRIC POLYNOMIALS ALEJANDRO ADEM AND ZINOVY REICHSTEIN Abstract. The cohomology of the classifying space BU(n) of the unitary group can be identified with the the

More information

Notes 10: Consequences of Eli Cartan s theorem.

Notes 10: Consequences of Eli Cartan s theorem. Notes 10: Consequences of Eli Cartan s theorem. Version 0.00 with misprints, The are a few obvious, but important consequences of the theorem of Eli Cartan on the maximal tori. The first one is the observation

More information

Qualifying Exam Syllabus and Transcript

Qualifying Exam Syllabus and Transcript Qualifying Exam Syllabus and Transcript Qiaochu Yuan December 6, 2013 Committee: Martin Olsson (chair), David Nadler, Mariusz Wodzicki, Ori Ganor (outside member) Major Topic: Lie Algebras (Algebra) Basic

More information

William G. Dwyer Clarence W. Wilkerson

William G. Dwyer Clarence W. Wilkerson Maps of BZ/pZ to BG William G. Dwyer Clarence W. Wilkerson The purpose of this note is to give an elementary proof of a special case of the result of [Adams Lannes 2 Miller-Wilkerson] characterizing homotopy

More information

Cover Page. Author: Yan, Qijun Title: Adapted deformations and the Ekedahl-Oort stratifications of Shimura varieties Date:

Cover 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 information

Applications of geometry to modular representation theory. Julia Pevtsova University of Washington, Seattle

Applications of geometry to modular representation theory. Julia Pevtsova University of Washington, Seattle Applications of geometry to modular representation theory Julia Pevtsova University of Washington, Seattle October 25, 2014 G - finite group, k - field. Study Representation theory of G over the field

More information

A Primer on Homological Algebra

A Primer on Homological Algebra A Primer on Homological Algebra Henry Y Chan July 12, 213 1 Modules For people who have taken the algebra sequence, you can pretty much skip the first section Before telling you what a module is, you probably

More information

Equivariant Algebraic K-Theory

Equivariant Algebraic K-Theory Equivariant Algebraic K-Theory Ryan Mickler E-mail: mickler.r@husky.neu.edu Abstract: Notes from lectures given during the MIT/NEU Graduate Seminar on Nakajima Quiver Varieties, Spring 2015 Contents 1

More information

Specialized Macdonald polynomials, quantum K-theory, and Kirillov-Reshetikhin crystals

Specialized 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 information

Realization problems in algebraic topology

Realization problems in algebraic topology Realization problems in algebraic topology Martin Frankland Universität Osnabrück Adam Mickiewicz University in Poznań Geometry and Topology Seminar June 2, 2017 Martin Frankland (Osnabrück) Realization

More information

arxiv:math/ v2 [math.at] 10 Mar 2007

arxiv:math/ v2 [math.at] 10 Mar 2007 arxiv:math/070113v [math.at] 10 Mar 007 THE INTEGRAL HOMOLOGY OF THE BASED LOOP SPACE ON A FLAG MANIFOLD JELENA GRBIĆ AND SVJETLANA TERZIĆ Abstract. The homology of a special family of homogeneous spaces,

More information

Spherical varieties and arc spaces

Spherical 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 information

BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n)

BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n) BLOCKS IN THE CATEGORY OF FINITE-DIMENSIONAL REPRESENTATIONS OF gl(m n) VERA SERGANOVA Abstract. We decompose the category of finite-dimensional gl (m n)- modules into the direct sum of blocks, show that

More information

Peter Magyar Research Summary

Peter Magyar Research Summary Peter Magyar Research Summary 1993 2005 Nutshell Version 1. Borel-Weil theorem for configuration varieties and Schur modules One of the most useful constructions of algebra is the Schur module S λ, an

More information

Primitive Ideals and Unitarity

Primitive Ideals and Unitarity Primitive Ideals and Unitarity Dan Barbasch June 2006 1 The Unitary Dual NOTATION. G is the rational points over F = R or a p-adic field, of a linear connected reductive group. A representation (π, H)

More information

Math 530 Lecture Notes. Xi Chen

Math 530 Lecture Notes. Xi Chen Math 530 Lecture Notes Xi Chen 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA E-mail address: xichen@math.ualberta.ca 1991 Mathematics Subject Classification. Primary

More information

Crossed products of irrational rotation algebras by finite subgroups of SL 2 (Z) are AF

Crossed products of irrational rotation algebras by finite subgroups of SL 2 (Z) are AF Crossed products of irrational rotation algebras by finite subgroups of SL 2 (Z) are AF N. Christopher Phillips 7 May 2008 N. Christopher Phillips () C (Z k, A θ ) is AF 7 May 2008 1 / 36 The Sixth Annual

More information

A relative version of Kostant s theorem

A 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 information

(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d

(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers

More information

Finer rook equivalence: Classifying Ding s Schubert varieties

Finer rook equivalence: Classifying Ding s Schubert varieties Finer rook equivalence: Classifying Ding s Schubert varieties Mike Develin Jeremy Martin Victor Reiner (AIM) (University of Minnesota) (University of Minnesota Preprint: arxiv:math.ag/4353 math.umn.edu/

More information

UC Berkeley Summer Undergraduate Research Program 2015 July 9 Lecture

UC Berkeley Summer Undergraduate Research Program 2015 July 9 Lecture UC Berkeley Summer Undergraduate Research Program 205 July 9 Lecture We will introduce the basic structure and representation theory of the symplectic group Sp(V ). Basics Fix a nondegenerate, alternating

More information

Longest element of a finite Coxeter group

Longest element of a finite Coxeter group Longest element of a finite Coxeter group September 10, 2015 Here we draw together some well-known properties of the (unique) longest element w in a finite Coxeter group W, with reference to theorems and

More information

The 3-primary Arf-Kervaire invariant problem University of Virginia

The 3-primary Arf-Kervaire invariant problem University of Virginia The 3-primary Arf-Kervaire invariant problem Mike Hill Mike Hopkins Doug Ravenel University of Virginia Harvard University University of Rochester Banff Workshop on Algebraic K-Theory and Equivariant Homotopy

More information

Representations of semisimple Lie algebras

Representations of semisimple Lie algebras Chapter 14 Representations of semisimple Lie algebras In this chapter we study a special type of representations of semisimple Lie algberas: the so called highest weight representations. In particular

More information

Weyl group representations on zero weight spaces

Weyl group representations on zero weight spaces Weyl group representations on zero weight spaces November 30, 2014 Here we survey briefly (trying to provide reasonably complete references) the scattered work over four decades most relevant to the indicated

More information

ABEL S THEOREM BEN DRIBUS

ABEL S THEOREM BEN DRIBUS ABEL S THEOREM BEN DRIBUS Abstract. Abel s Theorem is a classical result in the theory of Riemann surfaces. Important in its own right, Abel s Theorem and related ideas generalize to shed light on subjects

More information

UC Berkeley Summer Undergraduate Research Program 2015 July 8 Lecture

UC Berkeley Summer Undergraduate Research Program 2015 July 8 Lecture UC Berkeley Summer Undergraduate Research Program 25 July 8 Lecture This lecture is intended to tie up some (potential) loose ends that we have encountered on the road during the past couple of weeks We

More information

arxiv:math/ v2 [math.qa] 12 Jun 2004

arxiv: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 information

Division Algebras and Parallelizable Spheres, Part II

Division Algebras and Parallelizable Spheres, Part II Division Algebras and Parallelizable Spheres, Part II Seminartalk by Jerome Wettstein April 5, 2018 1 A quick Recap of Part I We are working on proving the following Theorem: Theorem 1.1. The following

More information

Geometric realization of PRV components and the Littlewood Richardson cone

Geometric 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 information

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and

More information

Topics in Representation Theory: SU(n), Weyl Chambers and the Diagram of a Group

Topics in Representation Theory: SU(n), Weyl Chambers and the Diagram of a Group Topics in Representation Theory: SU(n), Weyl hambers and the Diagram of a Group 1 Another Example: G = SU(n) Last time we began analyzing how the maximal torus T of G acts on the adjoint representation,

More information

REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE: EXERCISES. Notation. 1. GL n

REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE: EXERCISES. Notation. 1. GL n REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE: EXERCISES ZHIWEI YUN Fix a prime number p and a power q of p. k = F q ; k d = F q d. ν n means ν is a partition of n. Notation Conjugacy classes 1. GL n 1.1.

More information

Chern numbers and Hilbert Modular Varieties

Chern numbers and Hilbert Modular Varieties Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point

More information

Branching rules of unitary representations: Examples and applications to automorphic forms.

Branching rules of unitary representations: Examples and applications to automorphic forms. Branching rules of unitary representations: Examples and applications to automorphic forms. Basic Notions: Jerusalem, March 2010 Birgit Speh Cornell University 1 Let G be a group and V a vector space.

More information

7.3 Singular Homology Groups

7.3 Singular Homology Groups 184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the

More information

ON SOME PARTITIONS OF A FLAG MANIFOLD. G. Lusztig

ON 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 information

COMBINATORIAL MODELS FOR THE VARIETY OF COMPLETE QUADRICS

COMBINATORIAL 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 information

On the Rothenberg Steenrod spectral sequence for the mod 3 cohomology of the classifying space of the exceptional Lie group E 8

On the Rothenberg Steenrod spectral sequence for the mod 3 cohomology of the classifying space of the exceptional Lie group E 8 213 226 213 arxiv version: fonts, pagination and layout may vary from GTM published version On the Rothenberg Steenrod spectral sequence for the mod 3 cohomology of the classifying space of the exceptional

More information