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Nội dung bài viết là nêu và phân tích giải BGG của các biểu diễn bất khả quy của các nhóm tuyến tính tổng quát GLc(n). Mời các bạn tham khảo! | 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 Si xøng 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 GLn (C). biÓu diÔn bÊt kh¶ qui cña 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 T ⊆ GLn (C) relative to the maximal torus with the character GLn (C) , of T . In the Grothendieck Vλ the equivalence class of symmetric powers Sr (V ) of the [Vλ ] identity. Explicity, the class can GLn (C) ring of the be expressed as standard representation is the determinant of the [Sλi −i+j (V )] , keeping in mind that λ = (λ1 , λ2 , · · · , λn ) Z ⊆n of highest weigh of diagonal matrix, under the usual identification of S0 (V ) = C and that caterogy of polynomial a polynomial V = Cn n×n Sr (V ) = 0 of in the classes of GLn (C) (i, j) r<0 by -matrix whose for representations the of the various Jacobi - Trudi th entry is the class .[1] There is a useful formulation of the Jacobi - Trudi identity which utilizes the twisted dot action of the Weyl group GLn (C) GLn (C) on weights. representation of Πn of permutations of the V . set The Weyl group and permutation of coordinates. the dot action of where w ∈ W, λ ∈ Z n , we let denote the formed by taking the tensor product appropriate symmetric powers of group γ = (γ1 , γ2 , · · · , γn ) ∈ Z n S(γ Sγ1 (V ) ⊗ Sγ2 (V ) ⊗ · · · ⊗ Sγn (V ) W GLn (C) {1, 2, · · · ,
