By Laurent Manivel, John R. Swallow

This article grew out of a sophisticated path taught through the writer at the Fourier Institute (Grenoble, France). It serves as an creation to the combinatorics of symmetric features, extra accurately to Schur and Schubert polynomials. additionally studied is the geometry of Grassmannians, flag forms, and particularly, their Schubert forms. This booklet examines profound connections that unite those matters. The booklet is split into 3 chapters. the 1st is dedicated to symmetric capabilities and particularly to Schur polynomials. those are polynomials with confident integer coefficients within which every one of the monomials correspond to a tender tableau with the valuables of being "semistandard". the second one bankruptcy is dedicated to Schubert polynomials, which have been stumbled on by means of A. Lascoux and M.-P. Sch?tzenberger who deeply probed their combinatorial homes. it's proven, for instance, that those polynomials help the delicate connections among difficulties of enumeration of diminished decompositions of diversifications and the Littlewood-Richardson rule, a very efficacious model of that could be derived from those connections. the ultimate bankruptcy is geometric. it's dedicated to Schubert kinds, subvarieties of Grassmannians, and flag kinds outlined via sure occurrence stipulations with mounted subspaces. This quantity makes obtainable a couple of effects, making a good stepping stone for scaling extra bold heights within the zone. The author's rationale used to be to stay hassle-free: the 1st chapters require no previous wisdom, the 3rd bankruptcy makes use of a few rudimentary notions of topology and algebraic geometry. For this cause, a finished appendix at the topology of algebraic forms is supplied. This booklet is the English translation of a textual content formerly released in French.

**Read or Download Symmetric Functions, Schubert Polynomials and Degeneracy Loci (Smf Ams Texts and Monographs, Vol 6 and Cours Specialises Numero 3, 1998) PDF**

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**Extra resources for Symmetric Functions, Schubert Polynomials and Degeneracy Loci (Smf Ams Texts and Monographs, Vol 6 and Cours Specialises Numero 3, 1998)**

**Sample text**

Das Minimalpolynom f von α ist separabel und hat daher nur verschiedene Nullstellen. Es folgt direkt aus der Definition, dass D(f ) = 0. 7 folgt DF/K (1, α, α2 , . . 4 auch DF/K (β1 , . . , βn ) = 0. Wir wollen nun eine Methode zur Berechnung von DF/K (1, α, . . , αn−1 ) angeben, f¨ ur die man α nicht explizit zu kennen braucht. 10 Satz: Sei F = K(α), α separabel u ¨ber K n n−1 X + an−1 X + . . + a1 X + a0 . Dann gilt p0 p1 p1 p2 DF/K (1, α, . . , αn−1 ) = det .. . mit Minimalpolynom f = p2 p3 ..

N ) = det(σi (αj ))2 . Beweis: DF/K (α1 , . . , αn ) = det(SF/K (αi · αj )) = det( k σk (αi · αj )) = det( k σk (αi ) · σk (αj )) = det((σk (αi ))i,k · (σk (αj ))k,j ) = det(σk (αj ))2 . 6 Definition: Sei f ∈ K[X] ein Polynom vom Grad n und Leitkoeffizin ent c. Sei f = c · dann nennt man i=1 (X − αi ) seine Faktorisierung in Linearfaktoren in K[X], D(f ) = 1≤i

10 sind damit alle ci ganz u ¨ber R. Da ci ∈ K und R ganz abgeschlossen in K ist, sind alle ci ∈ R, also f ∈ R[X]. 18 Aufgabe: (1) Zeigen Sie: Sei R ganz abgeschlossener Integrit¨atsring und K sein Quotientenk¨orper. Sei f ∈ R[X] normiert und seien g, h ∈ K[X] normierte Polynome, so dass f =g·h Dann gilt g, h ∈ R[X]. 30 in K[X]. 17 mit Hilfe von (1). 19 Satz: Sei R ⊂ F ganze Ringerweiterung und F ein K¨orper. Dann ist auch R ein K¨orper. Beweis: Sei a = 0 aus R. Da a1 ∈ F ganz u ullt es eine Ganz¨ber R ist, erf¨ heitsgleichung mit Koeffizienten ci ∈ R 1 a n + cn−1 n−1 1 a + .