Introduction to the Theory of Standard Monomials: Second by C. S. Seshadri

By C. S. Seshadri

The e-book is a replica of a process lectures brought via the writer in 1983-84 which seemed within the Brandeis Lecture Notes sequence. the purpose of this path was once to provide an creation to the sequence of papers by means of targeting the case of the total linear workforce. in recent times, there was nice growth in regular monomial thought because of the paintings of Peter Littelmann. The author’s lectures (reproduced during this e-book) stay a very good advent to plain monomial theory.

Standard monomial conception bargains with the development of great bases of finite dimensional irreducible representations of semi-simple algebraic teams or, in geometric phrases, great bases of coordinate earrings of flag kinds (and their Schubert subvarieties) linked to those teams. in addition to its intrinsic curiosity, average monomial conception has purposes to the research of the geometry of Schubert forms. ordinary monomial idea has its beginning within the paintings of Hodge, giving bases of the coordinate jewelry of the Grassmannian and its Schubert subvarieties by way of “standard monomials”. In its smooth shape, commonplace monomial conception was once built via the writer in a sequence of papers written in collaboration with V. Lakshmibai and C. Musili. within the second edition of the e-book, conjectures of a typical monomial conception for a normal semi-simple (simply-connected) algebraic staff, as a result of Lakshmibai, were further as an appendix, and the bibliography has been revised.

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2 Lemma Ir be an element of the opposite big cell Uαmin . Let α ∈ A I(r, n) with α = αmin and let s be the integer such that αs ≤ r and αs+1 > r. If Let A = λ = (αs+1 , αs+2 , . . , αr ), µ = λ − (r) = (αs+1 − r, . . , αr − r), and ν = (1, 2, . . , α ˆ1 , α ˆ2 , . . , α ˆi, . . , α ˆ s , , αs + 1, . . , the complement of {α1 , α2 , . . , αs } in {1, 2, . . , r} arranged in increasing order), then pα (A) = ±pλ,ν (A) = ±pµ,ν (A). Proof. It is clear that the rows of the α-th minor of A which belong to Ir do not contribute any thing towards the value of pα (A) = ± det (α-th minor of A) and pα (A) is in fact determined by those rows of the α-th minor which belong to A.

Proof. Consider the homogeneous coordinate rings S = k[xα , α ∈ I(r, n)] and R = k[pα , α ∈ I(r, n)] of P(∧r ∨) and Gr,n respectively. We have a natural homomorphism Φ : S −→ R xα −→ pα whose kernel is the ideal J. 4) implies that J ′ ⊂ ker Φ = J. Hence we have a surjective homomorphism Φ : S/J ′ −→ R xα −→ pα . Thus, in order to prove that J ′ = J, it is enough to prove that Φ is injective. So let F be any nonzero element of S/J ′ . 9) S/J ′ is generated by standard monomials, and therefore F can be written as a linear combination of distinct standard monomials.

R ) then it is easy to see that γ = (γ1 , γ2 , . . , γr ), where γi = min(αi , βi ), 1 ≤ i ≤ r. 3). (iii) If µ = (µ1 , µ2 , . . , µr ) is a maximal element of Tα − {α}, it differs from α = (α1 , α2 , . . , αr ) just at one place by 1. 5,ii), the assertions in (iii) follow easily. To prove the remaining assertion, first we note that for any α in I(r, n), X(α) = X(Tα ). 8) and (i) above we have X(µ) ∩ X(µ′ ) = X(Tµ ) ∩ X(Tµ′ ) = X(Tµ ∩ Tµ′ ) = X(Tν ) = X(ν), where νi = min(µi , µ′i ), 1 ≤ i ≤ r.

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