By James A. Green, Manfred Schocker, Karin Erdmann (auth.)
ISBN-10: 3540469443
ISBN-13: 9783540469445
The first 1/2 this e-book comprises the textual content of the 1st variation of LNM quantity 830, Polynomial Representations of GLn. This vintage account of matrix representations, the Schur algebra, the modular representations of GLn, and connections with symmetric teams, has been the root of a lot study in illustration theory.
The moment part is an Appendix, and will be learn independently of the 1st. it's an account of the Littelmann direction version for the case gln. therefore, Littelmann's 'paths' develop into 'words', and so the Appendix works with the combinatorics on phrases. This results in the repesentation concept of the 'Littelmann algebra', that is a detailed analogue of the Schur algebra. The remedy is self- contained; specifically entire proofs are given of classical theorems of Schensted and Knuth.
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Example text
James describes, in his book [27], some KΓ-modules which are isomorphic to the Dλ,K . 8 that it yields the important and deep fact that Dλ,K is an “induced” module, in the sense of algebraic group theory. Chapter 6 returns to Schur’s dissertation. I have “reversed” the elegant procedure by which he constructed KΓ-modules from modules for the symmetric group G(r). This provides an interesting illumination of some recent work of James on the modular representation theory of the symmetric group. §2.
Is And for any field Sz(n,r) @ K ~ SK(n,r) which takes K ~ ,j ® IK ~ ~i,j The map e : KF ~ SK(n,r) For each g E F we define the element eg(C) = c(g) (c E AK(n,r)). 3b) that the multiplicatively closed - - i t K, GL ~(n,r) . 3a) and (i) that eI = s by the definition of linearly we get a map ~. e : KF ~ SK(n,r) egeg, = egg, So if we extend which a morphism K-algebras. Any function f : KF ~ K. e. e E ~(n,r). 4b,c) give the most important facts about e. 4b) Proposition (ii) Let Then f ( AK(n,r ) Proof (i) exist some (i) Y = Ker e, e is surjective.
We say that the family {θK } is defined over Z if θQ maps VZ into WZ , and for each K the diagram shown commutes. VZ ⊗ K θ Q ⊗ πK ηK δK VK G WZ ⊗ K θK G WK Example 2. Define the rth symmetric power Dr,K = Dr (EK ) of EK to be the rth homogeneous subspace of the polynomial ring K[e1 , . . , en ]; the elements e1 = e1,K , . . , en = en,K are regarded as commuting indeterminates. ⊗r → Dr (EK ) taking ei = ei1 ⊗ · · · ⊗ eir to There is a surjective K-map θK : EK the monomial e(i) = ei1 · · · eir , for all i ∈ I(n, r).
Polynomial Representations of GL n by James A. Green, Manfred Schocker, Karin Erdmann (auth.)
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