Abstract
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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Notes
- 1.
After the Swiss mathematician Eduard Stiefel, 21 April 1909–25 November 1978
- 2.
Two subvarieties V 1 and V 2 are said to be properly intersecting if the codimension of each irreducible component of \( V _{1} \cap V _{2} \) is the sum of the codimensions.
- 3.
The lack of space allowed by the book’s template prevented us from being more explicit, so we recommend the reader to follow the proofs of the cases r = 2, 3 to check that indeed the computations are correct and to sense what is going on.
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Gatto, L., Salehyan, P. (2016). Schubert Derivations. In: Hasse-Schmidt Derivations on Grassmann Algebras. IMPA Monographs, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-31842-4_5
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