Abstract
The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1; x2; : : :]. We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Gr¨obner geometry of matrix Schubert varieties.
Highlights
We wish to put forward another solution – a novel aspect of which is that it directly invokes semistandard tableaux. Both the statement and proof of our alternative model build upon ideas about the Grobner geometry of matrix Schubert varieties Xw
To prove the main theorem, we show there is a bijection between the set of prism tableau for w and Multi(w)
We identify each P ∈ MinPlus(Xw) with the prism tableau that corresponds to the minimum element of Multi(P) and show this is a bijection
Summary
The solutions [BiJoSt93, BeBi93, FoSt94, FoKi96] have been the foundation for a vast literature at the confluence of combinatorics, representation theory and combinatorial algebraic geometry. We wish to put forward another solution – a novel aspect of which is that it directly invokes semistandard tableaux. Both the statement and proof of our alternative model build upon ideas about the Grobner geometry of matrix Schubert varieties Xw. We use the Grobner degeneration of Xw and the interpretation of Sw as mutidegrees of Xw [KnMi05]. Is to establish the geometric naturality of the combinatorics of [BiJoSt93, BeBi93, FoKi96]. Knutson on Frobenius splitting [Kn09, Theorem 6 and Section 7.2]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Discrete Mathematics & Theoretical Computer Science
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
