Kevin Purbhoo

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The Belkale-Kumar product on H(G/P) is a degeneration of the usual cup product on the cohomology ring of a generalized flag manifold. In the case G = GLn, it was used by N. Ressayre to determine the regular faces of the Littlewood-Richardson cone. We show that forG/P a (d−1)-step flagmanifold, each Belkale-Kumar structure constant is a product of ( d 2 )(More)
The amoeba of an affine algebraic variety V ⊂ (C∗)r is the image of V under the map (z1, . . . , zr) 7→ (log |z1|, . . . , log |zr|). We give a characterisation of the amoeba based on the triangle inequality, which we call ‘testing for lopsidedness’. We show that if a point is outside the amoeba of V , there is an element of the defining ideal which(More)
For any complex reductive connected Lie group G, many of the structure constants of the ordinary cohomology ring H(G/B;Z) vanish in the Schubert basis, and the rest are strictly positive. We present a combinatorial game, the “root game”, which provides some criteria for determining which of the Schubert intersection numbers vanish. The definition of the(More)
We define mosaics, which are naturally in bijection with Knutson-Tao puzzles. We define an operation on mosaics, which shows they are also in bijection with Littlewood-Richardson skew-tableaux. Another consequence of this construction is that we obtain bijective proofs of commutativity and associativity for the ring structures defined either of these(More)
Vanishing and Non-Vanishing Criteria for Branching Schubert Calculus by Kevin Purbhoo Doctor of Philosophy in Mathematics University of California at Berkeley Professor Allen Knutson, Chair We investigate several related vanishing problems in Schubert calculus. First we consider the multiplication problem. For any complex reductive Lie group G, many of the(More)