Alladi Ramakrishnan Hall

A Groebner basis theorem for matrix Schubert varieties

K N Raghavan

IMSc

This talk has no prerequisites (should be accessible to all math doctoral students). It does not logically depend upon the talk from a fortnight ago, but is nevertheless a continuation of it.

We will recall the following two notions and the relation between them: (Stanley-Reisner) face ring of a simplicial complex;  Groebner basis of an ideal in a polynomial ring. We will then state the theorem due to Knutson-Miller giving a Groebner basis for the ideal of functions vanishing on the matrix Schubert variety associated to a permutation (whose definition we will recall). The maximal faces of the resulting simplicial complex are parametrized by combinatorial objects called reduced pipe dreams,  which are a generalization of semi-standard Young tableaux (for a Grassmannian permutation,  reduced pipe dreams are nothing but semi-standard Young tableaux).

Given that the Schubert polynomial associated to a permutation is realized as the multi-degree (with respect to a certain grading) of the ideal of the corresponding matrix Schubert variety,    one can recover, as a corollary of the above Groebner basis theorem,  combinatorial positive formulas (which were known earlier) for the coefficients of the Schubert polynomial.   As Knutson-Miller jokingly remark,   it is anybody's guess whether this recovery is a mere coincidence or a manifestation of the naturality of combinatorics.