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- Kevin Purbhoo
- 2009

The Wronskian associates to d linearly independent polynomials of degree at most n, a non-zero polynomial of degree at most d(n−d). This can be viewed as giving a flat, finite morphism from the Grassmannian Gr(d, n) to projective space of the same dimension. In this paper, we study the monodromy groupoid of this map. When the roots of the Wronskian are… (More)

The necessary and sufficient Horn inequalities which determine the nonvanishing Littlewood-Richardson coefficients in the cohomology of a Grassmannian are recursive in that they are naturally indexed by non-vanishing Littlewood-Richardson coefficients on smaller Grassmannians. We show how non-vanishing in the Schubert calculus for cominuscule flag varieties… (More)

- Allen Knutson, Kevin Purbhoo
- Electr. J. Comb.
- 2011

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)

- Kevin Purbhoo
- 2004

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)

- Kevin Purbhoo
- 2008

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)

- Kevin Purbhoo
- 2008

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)

We determine the necessary and sufficient combinatorial conditions for which the tensor product of two irreducible polynomial representations of GL(n, C) is isomorphic to another. As a consequence we discover families of LittlewoodRichardson coefficients that are non-zero, and a condition on Schur non-negativity.

We give a combinatorial rule for computing intersection numbers on a flag manifold which come from products of Schubert classes pulled back from Grassmannian projections. This rule generalizes the known rule for Grassmannians.

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)

- Kevin Purbhoo
- 2008

For any complex semisimple 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 root game is… (More)