DOCSLIB.ORG

Stability of Schur Functors*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Stability of Schur Functors* JOURNAL OF ALGEBRA 195, 233]240Ž. 1997 ARTICLE NO. JA977049 Stability of Schur Functors* Alexandre I. Kabanov² Department of Mathematics, Michigan State Uni¨ersity, Wells Hall, East Lansing, Michigan 48824-1027 View metadata, citation and similar papersCommunicated at core.ac.uk by Peter Littelmann brought to you by CORE provided by Elsevier - Publisher Connector Received September 5, 1996 SymplecticŽ. resp. orthogonal Schur functors are defined on a vector space V with a non-degenerate skew-symmetricŽ. resp. symmetric bilinear form. Each of these functors determines an irreducible representation of spŽ.ŽV resp. so Ž..V . We prove that if dim V is sufficiently large, then the splitting into irreducible components of the composition of a symplecticŽ. orthogonal Schur functor and an ordinary Schur functor applied to V is independent of dim V. We also prove the stability of the branching for the symplecticŽ. orthogonal Schur functors. Q 1997 Academic Press INTRODUCTION Con¨ention. All vector spaces, Lie algebras, and their representations Ž are considered over the complex numbers. Given a partition l s l1 G . l2 G ??? G lk ) 0 we say that k is the length of l, and we denote it by dlŽ. Ýk l l <<l . We call is1 i the weight of , and we denote it by . We also say that l is a partition of <<l Žcf.wx 3, I.1. Each partition l determines a Schur functor Sl which associates to a vector space V an irreducible slŽ.V -module SlV, and each irreducible representation of slŽ.V can be obtained in this way. This allows us to define a group homomorphism of representation rings t : R sl R sl Ž.nn12¨ Ž. whenever n n by sending the representation of sl corresponding to 12F n1 a partition l to the representation of sl corresponding to the same n2 * Research supported in part by an Alfred P. Sloan Doctoral Dissertation Fellowship. ² On leave until September 1997, Max-Planck-Institut furÈ Mathematik, Gottfried-Claren- Strasse 26, 53225, Bonn, Germany. E-mail address: [email protected]. 233 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved. 234 ALEXANDRE I. KABANOV partition. Using t one can compare the decomposition into irreducible components in RŽ.sln for different n. A functor F which associates to a vector space V a representation FVŽ. of slŽ.V is called stable if the decomposition of FV Ž.into irreducible components is independent of dim V provided that dim V is sufficiently Ž large. One can similarly define stability with respect to V12[ V and Ž. Ž.. Ž.Ž. sl V12[sl V . The stability of the functors SSmlV and S nV12[ V is a classical resultwx 1 . In this paper we prove similar stability results for the symplectic and orthogonal Schur functors. We first state the results in the symplectic case. Let V be a vector space with a non-degenerate skew-symmetric bilinear form. A symplectic Schur functor S² l: associates to V an irreducible representation of spŽ.V . As in the case of slŽ.V this allows us to compare the decomposition into irreducible components in RŽ.spn for different n. THEOREM Ž.cf. Theorem 7 . If 2<<mdŽ. l Fdim V, then the decomposi- tion of the spŽ.V -module SSm Ž² l:V . into irreducible components is indepen- dent of dim V. Let Vi, i s 1, 2, be vector spaces with non-degenerate skew-symmetric bilinear forms. Ž. Ž. THEOREM cf. Theorem 8 . The decomposition of S²n:V12[ Vasa Ž. Ž. Ž. sp V12[sp V -module does not depend on dim Vi, i s 1, 2, when 2dn Ä4 Ž. Fmin dim Vi, and it does not depend on dim V1 pro¨ided that 2dn F dim V12 when dim V is fixed. Now we state the results in the orthogonal case. They are identical to those in the symplectic case. Let W, W12, and W be vector spaces with S non-degenerate symmetric bilinear forms, and w lx be the orthogonal Schur functor associated to l. THEOREM. If 2<<mdŽ. l Fdim W, then the decomposition of the Ž.SS Ž . so W -module m w lxW into irreducible components is independent of dim W. HEOREM S Ž. Ž. T. The decomposition of wnx W12[ Wasaso W 1[ Ž. Ž. so W2 -module does not depend on dim Wi, i s 1, 2, pro¨ided 2dn F Ä4 Ž. min dim Vi, and it does not depend on dim W1 pro¨ided that 2dn F dim W12 when dim W is fixed. The only difference between the symplectic and the orthogonal cases is that in the latter case the orthogonal Schur functors describe a subring of the representation ring rather than the whole ring. However, all represen- tations which we consider in this note can be decomposed into irreducible STABILITY OF SCHUR FUNCTORS 235 components inside these subrings, and the difference above creates no difficulties. The proofs in the orthogonal case are identical to those in the symplectic case, and therefore will be omitted. The theorems above are a part of the mathematical folklore, but I am not aware of a relevant reference. The primary motivation for writing this paper is an article of Richard Hainwx 2, Sect. 6 . He uses the stability of the composition for the symplectic Schur functors in the study of the Torelli Lie algebras. This paper has the following structure. In the first two sections we recall relevant facts concerning stability in the representation rings of the special linear and symplectic Lie algebras. In the next two sections we use these facts to prove the stability of the compositionŽ. plethysm and branching for the symplectic Schur functors. 1. CLASSICAL SCHUR FUNCTORS For each partition l of d one can define the corresponding Young symmetrizer cl lying in the group ring of the symmetric group Sd wx1, p. 46 . The Schur functor Sl takes a vector space V to the image of cl: md md SlV s ImŽ.cl: V ª V . Let n be the dimension of V. Then the sln-modules SlV with dlŽ.-n are in one-to-one correspondence with the equivalence classes of irre- ducible representations of sln wx1, Proposition 15.15 . We denote by Ä4el , dlŽ.-n, the corresponding additive basis of RŽsl ..n . Now we list several stability properties of this representation ring. In order to do this we need to introduce a partial order on the set of all partitions. We say that m F l if miiF l for all i, and m - l if in addition mii-lfor some i. We also adopt a convention that the value of a functor on an element of a representation ring is determined by the value of this functor on the corresponding representation. PROPOSITION 1wx 1, Example 6.17; 3, I.8 . If <<mdŽ. l -n,then SmŽ.ells Ý Cemnn,1.1Ž. n:dnŽ.F<<mdŽ. l where Clmn are integers independent of n. Fix an inclusion sllm[sl ¨sl lqm. It induces the restriction homo- morphism r : RŽ.ŽsllqmlmªR sl [sl . 236 ALEXANDRE I. KABANOV XY commuting with the Schur functors. Let Äeel m4,dlŽ.-l,dm Ž .-m,bean Ž. additive basis of R sllm[sl whose elements correspond to irreducible representations. PROPOSITION 21,p.80.wxIf dnŽ.-minÄ4l, m , then X Y XY Ž.1.2 r Ž.ens Sn ŽeŽ1.Žq e 1. .s Ý Melmn le m , l, m: lFn , mFn where Mlmn are Littlewood]Richardson coefficients and they do not depend on l, m. If m is fixed and dnŽ.Fl,then X Y X X Y Ž.1.3 r Ž.ens Sn ŽeŽ1.Žq e 1. .s Ý Mlmne le m , l,m:lFn,dmŽ.-m X where Mlmn are integers independent of l. Another ingredient we will need is the transition from the basis Ä4el of RŽŽ..sl V to the basis corresponding to the products of the symmetric powers. We denote by SymlV the tensor product l l l Sym 1V m Sym 2 V m ??? m Sym k V, where l is a partition Ž.l12, l ,...,lk. We also introduce a lexicographic ordering on the set of partitions of the same weight: we say that m $ l if the first non-vanishing liiy m is positive. PROPOSITION 3wx 1, p. 86; 3, I.6.5 . There is an isomorphism of slŽ.V - modules SymlV S V K S V , ( l [ ž/[ ml m m:<<<<msl,m$l where Kml are non-negati¨e integers independent of dim V. This implies that the transition matrix Ž.Kml is unitriangular, and therefore invertible over the integers. 2. SYMPLECTIC SCHUR FUNCTORS Suppose that a vector space V of dimension n s 2 p carries a non- degenerate skew-symmetric form²: , . This form determines a contrac- d Žd 2. tion V m ª V m y for each two distinct factors of the tensor product. ² d: d Let V : V m be the intersection of the kernels of all these contrac- tions. STABILITY OF SCHUR FUNCTORS 237 DEFINITION 4wx 1, pp. 263]264 . The symplectic Schur functor S² l: associates to a pair Ž²:.V, , the sp2 p-module ² d: ² d: ² d: S²l:V s V l SllV s ImŽ.c : V ª V . The symplectic Schur functor S²l:V vanishes if and only if dlŽ.)p. There is a one-to-one correspondence between irreducible representations of sp2 p and partitions whose length is bounded by p wx1, Theorem 17.11 .
Load more

Recommended publications
  • Arxiv:1302.5859V2 [Math.RT] 13 May 2015 Upre Ynfflosi DMS-0902661
    STABILITY PATTERNS IN REPRESENTATION THEORY STEVEN V SAM AND ANDREW SNOWDEN Abstract. We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is an array of equivalences between the stable representation category and various other categories, each of which has its own flavor (representation theoretic, combinatorial, commutative algebraic, or categorical) and offers a distinct perspective on the stable category. We use this theory to produce a host of specific results, e.g., the construction of injective resolutions of simple objects, duality between the orthogonal and symplectic theories, a canonical derived auto-equivalence of the general linear theory, etc. Contents 1. Introduction 2 1.1. Overview 2 1.2. Descriptions of stable representation theory 4 1.3. Additional results, applications, and remarks 7 1.4. Relation to other work 10 1.5. Future directions 11 1.6. Organization 13 1.7. Notation and conventions 13 2. Preliminaries 14 2.1. Representations of categories 14 2.2. Polynomial representations of GL(∞) and category V 19 2.3. Twisted commutative algebras 22 2.4. Semi-group tca’s and diagram categories 24 2.5. Profinite vector spaces 25 arXiv:1302.5859v2 [math.RT] 13 May 2015 3. The general linear group 25 3.1. Representations of GL(∞) 25 3.2. The walled Brauer algebra and category 29 3.3. Modules over Sym(Ch1, 1i) 33 3.4. Rational Schur functors, universal property and specialization 36 4. The orthogonal and symplectic groups 39 4.1.
    [Show full text]
  • SCHUR-WEYL DUALITY Contents Introduction 1 1. Representation
    SCHUR-WEYL DUALITY JAMES STEVENS Contents Introduction 1 1. Representation Theory of Finite Groups 2 1.1. Preliminaries 2 1.2. Group Algebra 4 1.3. Character Theory 5 2. Irreducible Representations of the Symmetric Group 8 2.1. Specht Modules 8 2.2. Dimension Formulas 11 2.3. The RSK-Correspondence 12 3. Schur-Weyl Duality 13 3.1. Representations of Lie Groups and Lie Algebras 13 3.2. Schur-Weyl Duality for GL(V ) 15 3.3. Schur Functors and Algebraic Representations 16 3.4. Other Cases of Schur-Weyl Duality 17 Appendix A. Semisimple Algebras and Double Centralizer Theorem 19 Acknowledgments 20 References 21 Introduction. In this paper, we build up to one of the remarkable results in representation theory called Schur-Weyl Duality. It connects the irreducible rep- resentations of the symmetric group to irreducible algebraic representations of the general linear group of a complex vector space. We do so in three sections: (1) In Section 1, we develop some of the general theory of representations of finite groups. In particular, we have a subsection on character theory. We will see that the simple notion of a character has tremendous consequences that would be very difficult to show otherwise. Also, we introduce the group algebra which will be vital in Section 2. (2) In Section 2, we narrow our focus down to irreducible representations of the symmetric group. We will show that the irreducible representations of Sn up to isomorphism are in bijection with partitions of n via a construc- tion through certain elements of the group algebra.
    [Show full text]
  • The Partition Algebra and the Kronecker Product (Extended Abstract)
    FPSAC 2013, Paris, France DMTCS proc. (subm.), by the authors, 1–12 The partition algebra and the Kronecker product (Extended abstract) C. Bowman1yand M. De Visscher2 and R. Orellana3z 1Institut de Mathematiques´ de Jussieu, 175 rue du chevaleret, 75013, Paris 2Centre for Mathematical Science, City University London, Northampton Square, London, EC1V 0HB, England. 3 Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755, USA Abstract. We propose a new approach to study the Kronecker coefficients by using the Schur–Weyl duality between the symmetric group and the partition algebra. Resum´ e.´ Nous proposons une nouvelle approche pour l’etude´ des coefficients´ de Kronecker via la dualite´ entre le groupe symetrique´ et l’algebre` des partitions. Keywords: Kronecker coefficients, tensor product, partition algebra, representations of the symmetric group 1 Introduction A fundamental problem in the representation theory of the symmetric group is to describe the coeffi- cients in the decomposition of the tensor product of two Specht modules. These coefficients are known in the literature as the Kronecker coefficients. Finding a formula or combinatorial interpretation for these coefficients has been described by Richard Stanley as ‘one of the main problems in the combinatorial rep- resentation theory of the symmetric group’. This question has received the attention of Littlewood [Lit58], James [JK81, Chapter 2.9], Lascoux [Las80], Thibon [Thi91], Garsia and Remmel [GR85], Kleshchev and Bessenrodt [BK99] amongst others and yet a combinatorial solution has remained beyond reach for over a hundred years. Murnaghan discovered an amazing limiting phenomenon satisfied by the Kronecker coefficients; as we increase the length of the first row of the indexing partitions the sequence of Kronecker coefficients obtained stabilises.
    [Show full text]
  • Boolean Product Polynomials, Schur Positivity, and Chern Plethysm
    BOOLEAN PRODUCT POLYNOMIALS, SCHUR POSITIVITY, AND CHERN PLETHYSM SARA C. BILLEY, BRENDON RHOADES, AND VASU TEWARI Abstract. Let k ≤ n be positive integers, and let Xn = (x1; : : : ; xn) be a list of n variables. P The Boolean product polynomial Bn;k(Xn) is the product of the linear forms i2S xi where S ranges over all k-element subsets of f1; 2; : : : ; ng. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call Chern plethysm. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials Bn;k(Xn) for certain k to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate Bn;n−1(Xn) to a bigraded action of the symmetric group Sn on a divergence free quotient of superspace. 1. Introduction The symmetric group Sn of permutations of [n] := f1; 2; : : : ; ng acts on the polynomial ring C[Xn] := C[x1; : : : ; xn] by variable permutation. Elements of the invariant subring Sn (1.1) C[Xn] := fF (Xn) 2 C[Xn]: w:F (Xn) = F (Xn) for all w 2 Sn g are called symmetric polynomials. Symmetric polynomials are typically defined using sums of products of the variables x1; : : : ; xn. Examples include the power sum, the elementary symmetric polynomial, and the homogeneous symmetric polynomial which are (respectively) (1.2) k k X X pk(Xn) = x1 + ··· + xn; ek(Xn) = xi1 ··· xik ; hk(Xn) = xi1 ··· xik : 1≤i1<···<ik≤n 1≤i1≤···≤ik≤n Given a partition λ = (λ1 ≥ · · · ≥ λk > 0) with k ≤ n parts, we have the monomial symmetric polynomial X (1.3) m (X ) = xλ1 ··· xλk ; λ n i1 ik i1; : : : ; ik distinct as well as the Schur polynomial sλ(Xn) whose definition is recalled in Section 2.
    [Show full text]
  • Plethysm and Lattice Point Counting
    Plethysm and lattice point counting Thomas Kahle OvG Universit¨at Magdeburg joint work with M. Micha lek Some representation theory • Finite-dimensional, rational, irreducible representations of GL(n) are indexed by Young diagrams with at most n rows. • The correspondence is via the Schur functor: λ $ Sλ. • Sλ applied to the trivial representation gives an irreducible representation. Example and Convention • Only one row (λ = (d)): SλW = SdW (degree d forms on W ) • Only one column (λ = (1;:::; 1)): Sλ(W ) = Vd W Schur functor plethysm The general plethysm A Schur functor Sµ can be applied to any representation, for instance an irreducible SνW µ ν L λ ⊕cλ • S (S W ) decomposes into λ(S ) . • General plethysm: Determine cλ as a function of (λ, µ, ν). ! impossible? More modest goals { Symmetric powers • Decompose Sd(SkW ) into irreducibles. ! still hard. • Low degree: Fix d, seek function of (k; λ). Proposition (Thrall, 1942) One has GL(W )-module decompositions 2 M ^ M S2(SkW ) = SλW; (SkW ) = SµW; where • λ runs over tableaux with 2k boxes in two rows of even length. • µ runs over tableaux with 2k boxes in two rows of odd length. • Note: Divisibility conditions on the appearing tableaux. • Similar formulas for S3(Sk) obtained by Agaoka, Chen, Duncan, Foulkes, Garsia, Howe, Plunkett, Remmel, Thrall, . • A few things are known about S4(Sk) (Duncan, Foulkes, Howe). • Observation: Tableaux counting formulas get unwieldy quickly. Theorem (KM) λ d k Fix d. For any k 2 N, and λ ` dk, the multiplicity of S in S (S ) is d−1 X Dα X (−1)( 2 ) sgn(π)Q (k; λ ) ; d! α π α`d π2Sd−1 where • Qα are counting functions of parametric lattice polytopes • Dα is the number of permutations of cycle type α.
    [Show full text]
  • Topics in Group Representation Theory
    Topics in Group Representation Theory Peter Webb October 2, 2017 Contents 0.1 Representations of Sr ............................1 0.2 Polynomial representations of GLn(k)...................1 0.3 Representations of GL(Fpn ).........................2 0.4 Functorial methods . .2 1 Dual spaces and bilinear forms 3 2 Representations of Sr 4 2.1 Tableaux and tabloids . .4 2.2 Permutation modules . .5 2.2.1 Exercise . .7 2.3 p-regular partitions . .7 2.4 Young modules . .9 3 The Schur algebra 11 3.1 Tensor space . 11 3.2 Homomorphisms between permutation modules . 14 3.3 The structure of endomorphism rings . 15 3.4 The radical . 17 3.5 Projective covers, Nakayama's lemma and lifting of idempotents . 20 3.6 Projective modules for finite dimensional algebras . 24 3.7 Cartan invariants . 26 3.8 Projective and simple modules for SF (n; r)................ 28 3.9 Duality and the Schur algebra . 30 4 Polynomial representations of GLn(F ) 33 4.0.1 Multi-indices . 39 4.1 Weights and Characters . 40 5 Connections between the Schur algebra and the symmetric group: the Schur functor 47 5.1 The general theory of the functor f : B -mod ! eBe -mod . 51 5.2 Applications . 51 i CONTENTS ii 6 Representations of the category of vector spaces 54 6.1 Simple representations of the matrix monoid . 55 6.2 Simple representations of categories and the category algebra . 57 6.3 Parametrization of simple and projective representations . 60 6.4 Projective functors . 63 7 Bibliography 68 Landmark Theorems 0.1 Representations of Sr λ Theorem 0.1.1.
    [Show full text]
  • Schur Functors in This Lecture We Will Examine the Schur Polynomials and Young Tableaux to Obtain Irreducible Representations of Sl(N +1, C)
    MATH 223A NOTES 2011 LIE ALGEBRAS 91 23. Schur functors In this lecture we will examine the Schur polynomials and Young tableaux to obtain irreducible representations of sl(n +1, C). The construction is based on Fulton’s book “Young Tableaux” which is highly recommended for beginning students because it is a beautifully written book with minimal prerequisites. This will be only in the case of L = sl(n +1,F). Other cases and the proof of the Weyl character formula will be next. First, we construct the irreducible representations V (λi) corresponding to the funda- mental dominant weights λi = 1 + + i. Then we use Young tableaux to find the modules V (λ) inside tensor products··· of these fundamental representations. 23.1. Fundamental representations. Theorem 23.1.1. The fundamental representation V (λj) is equal to the ith exterior power of V = F n+1: V (λ )= jV j ∧ L = sl(n +1,F) acts by the isomorphism sl(n +1,F) ∼= sl(V ). Definition 23.1.2. Recall that the jth exterior power of a vector space V is the quotient of the jth tensor power by all elements of the form w1 wj where the wi are not j ⊗···⊗ distinct. The image of w1 wj in V is denoted w1 wj. When the characteristic of F is not equal to 2, this⊗··· is equivalent⊗ ∧ to saying that∧ the··· wedge∧ is skew symmetric: w w = sgn(σ)w w σ(1) ∧···∧ σ(j) 1 ∧···∧ j If v , ,v is a basis for V then 1 ··· n+1 v v v i1 ∧ i2 ∧···∧ ij for 1 i <i < <i n + 1 is a basis for jV .
    [Show full text]
  • Generalized Bialgebras and Triples of Operads
    Astérisque JEAN-LOUIS LODAY Generalized bialgebras and triples of operads Astérisque, tome 320 (2008) <http://www.numdam.org/item?id=AST_2008__320__R1_0> © Société mathématique de France, 2008, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 320 ASTERISQUE 2008 GENERALIZED BIALGEBRAS AND TRIPLES OF OPERADS Jean-Louis Loday SOCIÉTÉ MATHÉMATIQUE DE FRANCE Publié avec le concours du CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE Jean-Louis Loday Institut de Recherche Mathématique Avancée, CNRS et Université de Strasbourg, 7 rue R. Descartes, 67084 Strasbourg (France) [email protected] www-irma.u-strasbg.fr/~loday/ Mathematical classification by subject (2000).— 16A24, 16W30, 17A30, 18D50, 81R60. Keywords. — Bialgebra, generalized bialgebra, Hopf algebra, Cartier-Milnor-Moore, Poincaré-Birkhoff- Witt, operad, prop, triple of operads, primitive part, dendriform algebra, duplicial algebra, pre-Lie algebra, Zinbiel algebra, magmatic algebra, tree, nonassociative algebra. Many thanks to Maria Ronco and to Bruno Vallette for numerous conversations on bialgebras and operads. Thanks to E. Burgunder, B. Presse, R. Holtkamp, Y. Lafont, and M. Livernet for their comments. A special thank to F. Goichot for his careful reading. This work has been partially supported by the "Agence Nationale de la Recherche". We thank the anonymous referee for his/her helpful corrections and comments.
    [Show full text]
  • Resolutions of Homogeneous Bundles on P
    Resolutions of homogeneous bundles on P2 Giorgio Ottaviani Elena Rubei 1 Introduction Homogeneous bundles on P2 = SL(3)/P can be described by representations of the parabolic subgroup P . In 1966 Ramanan proved that if ρ is an irreducible 2 representation of P then the induced bundle Eρ on P is simple and even stable (see [Ram]). Since P is not a reductive group, there is a lot of indecomposable reducible representations of P and to classify homogeneous bundles on P2 and among them the simple ones, the stable ones, etc. by means of the study of the representations of the parabolic subgroup P seems difficult. In this paper our point of view is to consider the minimal free resolution of the bundle. Our aim is to classify homogeneous vector bundles on P2 by means of their minimal resolutions. Precisely we observe that if E is a homogeneous vector bundle on P2 = P(V ) (V complex vector space of dimension 3) there exists a minimal free resolution of E 0 →⊕qO(−q) ⊗C Aq →⊕qO(−q) ⊗C Bq → E → 0 with SL(V )-invariant maps (Aq and Bq are SL(V )-representations) and we char- acterize completely the representations that can occur as Aq and Bq and the maps A → B that can occur (A := ⊕qO(−q) ⊗C Aq, B = ⊕qO(−q) ⊗C Bq). To state the theorem we need some notation. arXiv:math/0401405v2 [math.AG] 22 Apr 2005 Notation 1 Let q, r ∈ N; for every ρ ≥ p, let ϕρ,p be a fixed SL(V )-invariant p,q,r ρ,q,r nonzero map S V ⊗ OP(V )(p) → S V ⊗ OP(V )(ρ) (it is unique up to multiples) ′ p,q,r s.t.
    [Show full text]
  • Schur Functors (A Project for Class)
    Schur Functors (a project for class) R. Vandermolen 1 Introduction In this presentation we will be discussing the Schur functor. For a complex vector space V , the Schur functor gives many irreducible representations of GL(V ), and other important subgroups of GL(V ), it will not be the purpose of this presentation to give this deep of an explanation. We will be concerning ourselves with the basic concepts needed to understand the Schur functor, and finish off by showing that the Schur functor does indeed help us decompose the nth tensor products V ⊗n. We will, for the most part, follow the presentation given in [1], and in turn we will use German gothic font for the symmetric group, and use diagrams in many occasions. With this said, we will not reprove major theorems from [1], instead we will focus our efforts on “filling in" the details of the presentation. We create new lemmas and deliver proofs of these new lemmas in efforts to deliver a fuller exposition on these Schur functors. Before we can begin a discussion of the Schur functor (or Weyl's module), we will need a familiarity with multilinear algebra. 1.1 Preliminaries Now we will deliver the definitions that will be of most interest to us in this presentation. We begin with the definition of the n-th tensor product. Definition 1. For a finite dimensional complex vector space V , and an natural number n, we define the complex n-th tensor product as follows V ⊗n := F (V n)= ∼ where F (V n) is the free abelian group on V n, considered as a set, and ∼ is the equivalence relation defined for v1; :::; vn; v^1; :::; v^n; w1; :::; wn 2 V , as (v1; :::; vn) ∼ (w1; :::; wn) whenever there exists a k 2 C such that k(v1; ::; vn) = (w1; :::; wn), and (v1; ::; vi; :::vn) + (v1; :::; v^i; :::; vn) ∼ (v1; ::; vi +v ^i; ::; vn) One can easily verify that this is indeed an equivalence relation.
    [Show full text]
  • Arxiv:1705.07604V2 [Math.CO] 7 Aug 2017 the Irreducible Representation of the General Linear Group Glm with the Highest Weight Λ
    SKEW HOWE DUALITY AND RANDOM RECTANGULAR YOUNG TABLEAUX GRETA PANOVA AND PIOTR SNIADY´ ABSTRACT. We consider the decomposition into irreducible compo- p m n nents of the external power Λ (C ⊗ C ) regarded as a GLm × GLn- module. Skew Howe duality implies that the Young diagrams from each pair (λ, µ) which contributes to this decomposition turn out to be con- jugate to each other, i.e. µ = λ0. We show that the Young diagram λ which corresponds to a randomly selected irreducible component (λ, λ0) has the same distribution as the Young diagram which consists of the boxes with entries ≤ p of a random Young tableau of rectangular shape with m rows and n columns. This observation allows treatment of the asymptotic version of this decomposition in the limit as m; n; p ! 1 tend to infinity. 1. INTRODUCTION 1.1. The problem. In this note we address the following question. Consider p m n M λ m λ0 n (1) Λ (C ⊗ C ) = S C ⊗ S C λ as a GLm × GLn-module. [. ] I would like any information on the shapes of pairs of Young diagrams (λ, λ0) that give the largest contribution to the dimension asymptotically. [. ] Joseph M. Landsberg [Lan12]1 Above, SλCm denotes the Schur functor applied to Cm or, in other words, arXiv:1705.07604v2 [math.CO] 7 Aug 2017 the irreducible representation of the general linear group GLm with the highest weight λ. The sum in (1) runs over Young diagrams λ ⊆ nm with p boxes, and such that the number of rows of λ is bounded from above by m, and the number of columns of λ is bounded from above by n.
    [Show full text]
  • SOME GENERALIZATIONS of SCHUR FUNCTORS 1. Introduction
    SOME GENERALIZATIONS OF SCHUR FUNCTORS STEVEN V SAM AND ANDREW SNOWDEN Abstract. The theory of Schur functors provides a powerful and elegant approach to the representation theory of GLn—at least to the so-called polynomial representations— especially to questions about how the theory varies with n. We develop parallel theories that apply to other classical groups and to non-polynomial representations of GLn. These theories can also be viewed as linear analogs of the theory of FI-modules. 1. Introduction The theory of Schur functors is a major achievement of representation theory. One can use these functors to construct the irreducible representations of GLn—at least those whose highest weight is positive (the so-called “polynomial” representations). This point of view is important because it gives a clear picture of how the irreducible representation of GLn with highest weight λ varies with n, which is crucial to understand when studying problems in which n varies. Unfortunately, this theory only applies to GLn and not other classical groups, and, as mentioned, only to the polynomial representations of GLn. The purpose of this paper is to develop parallel theories that apply in other cases. To explain our results, we focus on the orthogonal case. Let C = CO be the category whose objects are finite dimensional complex vector spaces equipped with non-degenerate symmetric bilinear forms and whose morphisms are linear isometries. An orthogonal Schur functor is a functor from C to the category of vector spaces that is algebraic, in an appro- priate sense. We denote the category of such functors by A = AO.
    [Show full text]