The BGG resolutions of irreducible representations of the general linear group GLn(C) - Nguyen Thi Phuong Dung

THE BGG RESOLUTIONS OF IRREDUCIBLE REPRESENTATIONS OF THE GENERAL LINEAR GROUP GLn(C) Nguyen Thi Phuong Dung Banking Academy Tãm t¾t: Ph©n lo¹i c¸c biÓu diÔn bÊt kh¶ qui cña nhãm tuyÕn tÝnh tæng qu¸t GLC(n) ®· hoµn toµn ®uîc gi¶i quyÕt. Trong ®ã ®Æc trung cña c¸c biÓu diÔn cã c«ng thøc m« t¶ rÊt ®Ñp ®Ï th«ng qua ®Þnh thuc cña c¸c ten s¬ ®èi xøng Si cña kh«ng gian vÐc t¬ V cè ®Þnh. Môc ®Ých bµi b¸o nµy lµ miªu t¶ cô thÓ viÖc x©y dùng phuc, mµ th«ng qua ®Æc trung Eueler - Poincare cña phøc khíp nµy, ta m« t¶ ®uîc c«ng thøc ®Þnh thøc tæng qu¸t cña c¸c biÓu diÔn bÊt kh¶ qui cña GLn(C). Tõ khãa: Gi¶i BGG, §Æc trung Eucler - Poincare, nhãm tuyÕn tÝnh tæng qu¸t, biÓu diÔn bÊt kh¶ qui, biÓu diÔn ®a thøc. 1 Introduction Let Vλ be the irreducible polynomial representation of GLn(C) of highest weigh λ = (λ1, λ2, · · · , λn) relative to the maximal torus T ⊆ GLn(C) of diagonal matrix, under the usual identification of Z ⊆n with the character of T . In the Grothendieck ring of the caterogy of polynomial representations of GLn(C), the equivalence class of Vλ can be expressed as a polynomial in the classes of the various symmetric powers Sr(V ) of the standard representation V = C n of GLn(C) by the Jacobi - Trudi identity. Explicity, the class [Vλ] is the determinant of the n× n-matrix whose (i, j)th entry is the class [Sλi−i+j(V )], keeping in mind that S0(V ) = C and that Sr(V ) = 0 for r < 0.[1] There is a useful formulation of the Jacobi - Trudi identity which utilizes the twisted dot action of the Weyl group GLn(C) on weights. For any weight γ = (γ1, γ2, · · · , γn) ∈ Zn, we let S(γ denote the representation of GLn(C) formed by taking the tensor product Sγ1(V ) ⊗ Sγ2(V ) ⊗ · · · ⊗ Sγn(V ) of appropriate symmetric powers of V . The Weyl group W of GLn(C) is isomorphic to the symmetric group Πn of permutations of the set {1, 2, · · · , n} and acts on the additive group Zn of weights by permutation of coordinates. the dot action ofW on Zn is defined by the twisted rulew◦λ = w(λ+δ)−δ, where w ∈ W,λ ∈ Zn, and δ denotes the sequence (n− 1, n− 2, · · · , 2, 1, 0). With these definitions, the Jacobi - Trudi identity can be written as a summation [Vλ] = Σwsign(w)[S(w · λ)] (1) over all permutations w ∈ W . For any nondecreasing nonnegative sequence λ ∈ Zn we will construct a resolution 0 −→ YCn2 (λ) −→ · · · −→ Y1(λ) −→ Y0(λ) −→ Vλ −→ 0 over GLn(C) which realizes the identity in (1) in the sense that each Yi(λ) is the direct sum ΣS(w ·λ) over all permutations w of length l(w) = i.[5] 1 *Tel: 0976605305, e-mail: phuongdung72@yahoo.com Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên 2 The BGG resolution of irreducible polynomial representations ofGLn(C) My purpose is to describe the BGG resolution for any irreducible polynomial representation of GLn(C) (i.e the representation with weight λ = (λ1, λ2, · · · , λn) ∈ Zn;λi ≥ λi+1. At first, we consider the BGG resolution of the trivial representation k of SL(n): C : 0 −→ CN −→ · · · −→ C1 −→−→ C0 −→ k −→ 0, where N = C2n and Ci = ΣMλ, Mλ is the Verma module, λ = w(δ) − δ, w ∈ Πn, l(w) = i, δ = (n− 1, n− 2, · · · , 1, 0). For any polynomial representation V of GLn(C), we consider that as a representation of sln(C), and the funtor φV,µ : M 7−→ [V ⊗M/n−](µ) where for any representation M ,M(µ) denoted subspace of M corresponding weight µ. φV,µ is a left exact funtor but not exact, however that change the above exact (C) to the exact sequence. More, Zelevinski proved that φV,µ(Mλφ) = Vµ−λ and φV,µ(Lλ) = Homsl(n)(Lµ, V ). So, the image of (C) by φV,µ is an exact complex, has form: 0→ C ′N → · · · → C ′1 → C ′0 → Homsl(n)(Lµ, V )→ 0 where C ′i = ⊕l(w)=iV(µ+δ−wδ). In general, this complex is only the exact sequence of the vector spaces. The Euler - Poincare function of this sequence gives the formula, which find the multiple of the simple module Lµ in V follow the dimension of weight spaces in V . The key point is the maps in the above complex don't depend on V representation and weight , only depend on permutations. More detail, the map V(µ+δ−wδ) 7−→ V(µ+δ−w′δ) only depends on w and w′. K. Akin want to detailed description these maps in [2]. 2.1 BGG resolution of the irirreducible representation of G = GL(2|1) The Resolution to describe the determinant formula of irreducible representations of GL(n) by the power of symmetry is trivial consequence of the complex C ′ above. Conider V as a representation of GL(n) on the d-th homogeneous component of the ring k[zji ]. In this case V also have an action of GL(n). Consider V as a bimodule,we have the Newton expansion V = ⊕λ 7→nLλ ⊗k Lλ Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên Therefore HomGL(n)(Lµ, V ) ∼= Lµ. On the other hand, consider V as GL(n) left module, we have V(λ) = ⊗Sλi as GL(n) module with right action above. Hence C ′ is the complex to find. Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên References [1] K. Akin. Resolutions of Representations. Contemp. Math., in press, 1986. [2] K. Akin and .D. A. Buchsbaum. Characteristic - free representation theory of the general linear group. Adv. in Math., 39, 1985, 149 - 200. [3] A. Berele and A. Regev. Hook Young Diagrams with Applications to Combinatorics and to Repre- sentation of Lie Algebras. Advances in Math., 64:118--175, 1987. [4] J.Brundan. Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra gl(m|n). J. Amer. Math. Soc., 16:185-231, 2002. [5] J. A. Green. Polynomial representations of GL(n) Lecture notes in Mathematics, Vol. 308, Spring - Verlag, Berlin, 1980. [6] J. E. Humpheys. Modula representations of finite groups of Lie type, Proc. of London Math. Soc, 1978, 259-290 [7] V.G.Kac. Classification of simple Lie superalgebras. Funct.Anal. Appl., 9:263-265, 1975. [8] V.G.Kac. Lie superalgebras. Adv. Math., 26:8-96, 1977. [9] V.G.Kac. Character of typical representations of classical Lie superalgebras. Comm. Alg., 5:889-897, 1977. [10] V.G.Kac. Representations of classical Lie superalgebras, . in: Lecture Notes in Math ., 676:597- 626, 1978. [11] I.G.Macdonald. Symmetric Function and the Hall Polynomials. Oxford University Press, Newyork,1979.. [12] M.Scheunert. The Theory of Lie Superalgebras. Lectune notes in Math., Springer-Verlag,1978 [13] M.Scheurt, R.B.Zhang. The general linear supergroup and its Hopf superalgebra of regular func- tions. Jour. Alg., 254:44-83, 2002. [14] Yucai Su, R.B.Zhang. Character and dimensoin formulae for general linear superalgebra. Adv. Math., 211:1-33, 2007. SUMMARY: The BGG resolution of irreducible representations of GL(n) The purpose of this paper is desecribe how the resolution of Bernstein - Gelfand - Gelfand (BBG) may be used to realize the Jacobi - Trudi expansion of a Schur function as a resolution in the catelogy of polynomial representations of the general linear group GLn(C) Nguyen Thi Phuong Dung Banking Academy 1 *Tel: 0976605305, e-mail: phuongdung72@yahoo.com Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên