BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:K. N. Raghavan (Institute of Mathematical Sciences\, Chennai) DTSTART:20201002T120000Z DTEND:20201002T130000Z DTSTAMP:20260423T021001Z UID:VCAS/26 DESCRIPTION:Title: Mu ltiplicities of points on Schubert varieties in the Grassmannian-II\nb y K. N. Raghavan (Institute of Mathematical Sciences\, Chennai) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nGiven an ar bitrary point on a Schubert (sub)variety in a Grassmannian\, how to comp ute the Hilbert function (and\, in particular\, the multiplicity) of the l ocal ring at that point? A solution to this problem based on "standard monomial theory" was conjectured by Kreiman-Lakshmibai circa 2000 and the conjecture was proved about a year or two later by them and independently also by Kodiyalam and the speaker. The two talks will be an exposition o f this material aimed at non-experts in the sense that we will not presume familiarity with Grassmannians (let alone flag varieties) or Schubert var ieties. \n\nThere are two steps to the solution. The first translate s the problem from geometry to algebra and in turn to combinatorics. The second is a solution of the resulting combinatorial problem\, which invo lves establishing a bijection between two combinatorially defined sets. The two talks will roughly deal with these two steps respectively.\n\nThr ee aspects of the combinatorial formulation of the problem (and its soluti on) are noteworthy: (A) it shows that the natural determinantal generat ors of the tangent cone (at the given point) form a Groebner basis (in any "anti-diagonal" term order)\; (B) it leads to an interpretation of the mu ltiplicity as counting certain non-intersecting lattice paths\; and (C) a s was observed by Kreiman some years later\, the combinatorial bijection is a kind of Robinson-Schensted-Knuth correspondence\, which he calls th e "bounded RSK".\n\nDeterminantal varieties arise as tangent cones at poin ts on Schubert varieties (in the Grassmannian)\, and thus one recovers mul tiplicity formulas for these obtained earlier by Abhyankar and Herzog-Trun g. (The multiplicity part of the Kreiman-Lakshmibai conjecture was also p roved by Krattenthaler\, but by very different methods.)\n\nWhat about Sc hubert varieties in other (full or partial) flag varieties (G/Q with Q bei ng a parabolic subgroup of a reductive algebraic group G)? The problem r emains open in general\, even for the case of the full flag variety GL(n)/ B\, although there are several papers over the last two decades by vario us authors using various methods that solve the problem in various special cases. Time permitting\, we will give some indication of these result s\, without however any attempt at comprehensiveness.\n LOCATION:https://researchseminars.org/talk/VCAS/26/ END:VEVENT END:VCALENDAR