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2016 | OriginalPaper | Buchkapitel

5. Schubert Derivations

verfasst von : Letterio Gatto, Parham Salehyan

Erschienen in: Hasse-Schmidt Derivations on Grassmann Algebras

Verlag: Springer International Publishing

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Let A be a commutative ring with unit, p ∈ A[X] a monic polynomial of degree n > 0, M _p the quotient of A[X] modulo (p) and \( b_{i}:= X^{i} + (p) \). The protagonist of this chapter is a distinguished HS-derivation on \( \bigwedge M_{p} \), called Schubert derivation. It is the unique HS-derivation \( \sigma _{+}(z):=\sum _{i\geq 0}\sigma _{i}z^{i} \) such that \( \sigma _{i}b_{j} = b_{i+j} \). The subscript ‘+’ of \( \sigma \) in the notation is to mark the difference from its relative \( \sigma _{-}(z):=\sum _{i\geq 0}\sigma _{-i}z^{-i} \), to be investigated in Chapter 6 Its connection with Schubert calculus, summarized in Section 5.4, where a few notions of intersection theory are outlined as well, motivates notation and terminology: it is based on Pieri- and Giambelli-type formulas enjoyed by the coefficients of the Schubert derivation. Pieri’s rule is treated in Section 5.2, in its quantum version as well, while Giambelli’s is proved in Section 5.8, basing on a flexible determinantal formula due to Laksov and Thorup as in [96, 97].

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Nächstes Kapitel Decomposable Tensors in Exterior Powers

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Titel: Schubert Derivations
verfasst von: Letterio Gatto
Parham Salehyan
Verlag: Springer International Publishing
Buch: Hasse-Schmidt Derivations on Grassmann Algebras
Print ISBN: 978-3-319-31841-7

Electronic ISBN: 978-3-319-31842-4

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-319-31842-4_5

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