Related papers: An $E_\infty$ structure on the matroid grassmannia…
A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering…
We define the notion of a relative matrad and realize the free relative matrad as a free H_\infty-bimodule structure on cellular chains of bimultiplihedra JJ={JJ_{n,m} = JJ_{m,n}}. We define a morphism G:A => B of A_\infty-bialgebras as a…
In this article a family of recursive and self-similar matrices is constructed. It is shown that the Pl\"ucker matrix of the Isotropic Grassmannian variety is a direct sum of this class of matrices.
We prove an explicit combinatorial formula for the structure constants of the Grothendieck ring of a Grassmann variety with respect to its basis of Schubert structure sheaves. We furthermore relate K-theory of Grassmannians to a bialgebra…
A structure M is pregeometric if the algebraic closure is a pregeometry in all M' elementarily equivalent to M. We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power…
We introduce the notion of a matrad M = {M_{n,m}} whose submodules M_{*,1} and M_{1,*} are non-Sigma operads. We define the free matrad H_{\infty} generated by a singleton in each bidegree (m,n) and realize H_{\infty} as the cellular chains…
The theory of matroids or combinatorial geometries originated in linear algebra and graph theory, and has deep connections with many other areas, including field theory, matching theory, submodular optimization, Lie combinatorics, and total…
The nonnegative Grassmannian is a cell complex with rich geometric, algebraic, and combinatorial structures. Its study involves interesting combinatorial objects, such as positroids and plabic graphs. Remarkably, the same combinatorial…
In the paper we prove an extension theorem for matrices with entries in H^{\infty}(U) for U being a Riemann surface of a special type. One of the main components of the proof is a Grauert type theorem for "holomorphic" vector bundles…
This paper is a direct generalization of Baker-Bowler theory to flag matroids, including its moduli interpretation as developed by Baker and the second author for matroids. More explicitly, we extend the notion of flag matroids to flag…
We prove that for the action of a finite constant group scheme, equivariant algebraic $K$-theory is represented by a colimit of Grassmannians in the equivariant motivic homotopy category. Using this result we show that the set of…
Patrias and Pylyavskyy introduced shifted Hecke insertion as an application of their theory of dual filtered graphs. We use shifted Hecke insertion to construct symmetric function representatives for the K-theory of the orthogonal…
We introduce the notion of lattice path matroidal subdivisions, or LPM subdivisions for short, and show that these subdivisions are regular and hence the weight vectors for them lie in the Dressian. This leads us to explore the structure of…
We consider the K-theory of the Hilbert scheme of points in the complex plane, which under McKay correspondence is isomorphic to the space of symmetric functions $\Lambda^n$. We prove a formula conjectured by Boissi\`ere for the…
In the present paper continuing our previous work we prove an extension theorem for matrices with entries in the algebra of bounded holomorphic functions defined on an unbranched covering of a Caratheodory hyperbolic Riemann surface of…
We construct a family of moduli spaces, called generalized Lagrangian flag schemes, that are reducible Gorenstein (hence singular), and that admit well-behaved pushforward and pullback operations in Hermitian $K$-theory. These schemes arise…
Delta-matroids are "type B" generalizations of matroids in the same way that maximal orthogonal Grassmannians are generalizations of Grassmannians. A delta-matroid analogue of the Tutte polynomial of a matroid is the interlace polynomial.…
We generalize the classical operad pair theory to a new model for $E_\infty$ ring spaces, which we call ring operad theory, and establish a connection with the classical operad pair theory, allowing the classical multiplicative infinite…
We introduce the notion of real phase structure on rational polyhedral fans in Euclidean space. Such a structure consists of an assignment of affine spaces over $\mathbb{Z}/2\mathbb{Z}$ to each top dimensional face of the fan subject to two…
We provide details of the proof of Lurie's theorem on operadic Kan extensions. Along the way, we generalize the construction of monoidal envelopes of $\infty$-operads to families of $\infty$-operads and use it to construct the fiberwise…