34 citations · 194 across the 17 of their papers we have counts for
18 papers
Totally nonnegative maximal tori and opposed Bruhat intervals
Grant T. Barkley, Steven N. Karp
Lusztig (2024) recently introduced the space of totally positive maximal tori of an algebraic group . Each such torus is the intersection of a totally positiv…
Zero-free sector of the Wronski map on the totally nonnegative Grassmannian
Steven N. Karp
A classical result states that if is a polynomial of degree at most with nonnegative coefficients, then has no zeros in the sector of the…
Richardson tableaux and components of Springer fibers equal to Richardson varieties
Steven N. Karp, Martha E. Precup
Motivated by the study of Springer fibers and their totally nonnegative counterparts, we define a new subset of standard tableaux called Richardson tableaux. We characterize Richar…
Positivity and universal Plücker coordinates for spaces of quasi-exponentials
Steven N. Karp, Evgeny Mukhin, Vitaly Tarasov
A quasi-exponential is an entire function of the form , where is a polynomial and . Let $V = \langle e^{h_1u}p_1(u), \dots, e^{h_Nu}p_N(u) \ran…
Universal Plücker coordinates for the Wronski map and positivity in real Schubert calculus
Steven N. Karp, Kevin Purbhoo
Given a -dimensional vector space of polynomials, its Wronskian is the polynomial whose zeros are the points of $\m…
Symmetric Toda, gradient flows, and tridiagonalization
Anthony M. Bloch, Steven N. Karp
The Toda lattice (1967) is a Hamiltonian system given by points on a line governed by an exponential potential. Flaschka (1974) showed that the Toda lattice is integrable by in…
q-Whittaker functions, finite fields, and Jordan forms
Steven N. Karp, Hugh Thomas
The -Whittaker function associated to a partition is a -analogue of the Schur function , and is defined as the specialization o…
On two notions of total positivity for partial flag varieties
Anthony M. Bloch, Steven N. Karp
Given integers , let denote the type partial flag variety consisting of all chains of subspaces $(V_{k_1}\subset…
Wronskians, total positivity, and real Schubert calculus
Steven N. Karp
A complete flag in is a sequence of nested subspaces such that each has dimension . It is called totally nonnegative if…
Gradient flows, adjoint orbits, and the topology of totally nonnegative flag varieties
Anthony M. Bloch, Steven N. Karp
One can view a partial flag variety in as an adjoint orbit inside the Lie algebra of skew-Hermitian matrices. We use the orbit context t…
Shelling the m=1 amplituhedron
Steven N. Karp, John Machacek
The amplituhedron was introduced by Arkani-Hamed and Trnka (2014) in order to give a geometric basis for calculating scattering amplitudes in planar $\mathcal…
Regularity theorem for totally nonnegative flag varieties
Pavel Galashin, Steven N. Karp, Thomas Lam
We show that the totally nonnegative part of a partial flag variety (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the…
Moment curves and cyclic symmetry for positive Grassmannians
Steven N. Karp
We show that for each k and n, the cyclic shift map on the complex Grassmannian Gr(k,n) has exactly fixed points. There is a unique totally nonnegative fixed point,…
The totally nonnegative part of G/P is a ball
Pavel Galashin, Steven N. Karp, Thomas Lam
We show that the totally nonnegative part of a partial flag variety (in the sense of Lusztig) is homeomorphic to a closed ball.
Decompositions of amplituhedra
Steven N. Karp, Lauren K. Williams, Yan X Zhang
The (tree) amplituhedron A(n,k,m) is the image in the Grassmannian Gr(k,k+m) of the totally nonnegative part of Gr(k,n), under a (map induced by a) linear map which is totally posi…
The totally nonnegative Grassmannian is a ball
Pavel Galashin, Steven N. Karp, Thomas Lam
We prove that three spaces of importance in topological combinatorics are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of e…
The m=1 amplituhedron and cyclic hyperplane arrangements
Steven N. Karp, Lauren K. Williams
The (tree) amplituhedron A(n,k,m) is the image in the Grassmannian Gr(k,k+m) of the totally nonnegative part of Gr(k,n), under a (map induced by a) linear map which is totally posi…
Sign variation, the Grassmannian, and total positivity
Steven N. Karp
The totally nonnegative Grassmannian is the set of k-dimensional subspaces V of R^n whose nonzero Pluecker coordinates all have the same sign. Gantmakher and Krein (1950) and Schoe…