Related papers: A representation-theoretic interpretation of posit…
We construct a complex of toric varieties we call the quasisymmetric Grassmannian inside the Grassmannian of $r$-planes in $\mathbb{C}^n$. Each irreducible component is a positroid variety and an $S_n$ translate of a toric Richardson…
In this paper we express certain multiplicities in modular representation-theoretic categories of type A in terms of affine p-Kazhdan-Lusztig polynomials. The representation-theoretic categories we deal with include the categories of…
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…
Motivated by questions about $\mathbb{C}_p$-valued Fourier transform on the locally compact group $(\mathbb{Q}_p^d,+)$, we study invariant norms on the $p$-adic Schr\"odinger representation of the Heisenberg group. Our main result is a…
We give a concrete combinatorial interpretation of the coefficients of the Kazhdan-Lusztig polynomials of Dowling geometries, a family of matroids which generalizes braid matroids of types A and B. Furthermore, we interpret the coefficients…
In the present paper, we provide a cohomology group as a categorification of the characteristic polynomial of matroids. The construction depends on the ``quasi-representation'' of a matroid. For a certain choice of the quasi-representation,…
We compute the expansion of the cohomology class of the permutahedral variety in the basis of Schubert classes. The resulting structure constants $a_w$ are expressed as a sum of \emph{normalized} mixed Eulerian numbers indexed naturally by…
A reciprocal linear space is the image of a linear space under coordinate-wise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path…
The Whitney module of a matroid is a natural analogue of the tensor algebra of the exterior algebra of a vector space that takes into account the dependencies of a matroid. In this paper we indicate the role that tableaux can play in…
It is well known that the Eulerian polynomial is the Hilbert series of the cohomology of the permutohedral variety. We answer a question of Stembridge on finding a geometric explanation of the \emph{permutation representation} this…
A bimatroid is a matroid-like generalization of the collection of regular minors of a matrix. In this article, we use the theory of Lorentzian polynomials to study the logarithmic concavity of natural sequences associated to bimatroids.…
Motivated by Kontsevich's graph complexes, this paper gives a systematic study of matroid complexes. We construct deletion and contraction bicomplexes on the vector space spanned by matroid classes equipped with ground-set orientations,…
Flag matroids are a rich family of Coxeter matroids that can be characterized using pairs of matroids that form a quotient. We consider a class of matroids called positroids, introduced by Postnikov, and utilize their combinatorial…
We provide two alternate settings for a family of varieties modeling the uniserial representations with fixed sequence of composition factors over a finite dimensional algebra. The first is a quasi-projective subvariety of a Grassmannian…
We study generalizations of Schur functors from categories consisting of flags of vector spaces. We give different descriptions of the category of such functors in terms of representations of certain combinatorial categories and infinite…
Schur modules give the irreducible polynomial representations of the general linear group $\mathrm{GL}_t$. Viewing the symmetric group $\mathfrak{S}_t$ as a subgroup of $\mathrm{GL}_t$, we may restrict Schur modules to $\mathfrak{S}_t$ and…
The classical matrix tree theorem relates the number of spanning trees of a connected graph with the product of the nonzero eigenvalues of its Laplacian matrix. The class of regular matroids generalizes that of graphical matroids, and a…
A rational Dyck path of type $(m,d)$ is an increasing unit-step lattice path from $(0,0)$ to $(m,d) \in \mathbb{Z}^2$ that never goes above the diagonal line $y = (d/m)x$. On the other hand, a positroid of rank $d$ on the ground set $[d+m]$…
Schubert varieties of hyperplane arrangements, also known as matroid Schubert varieties, play an essential role in the proof of the Dowling-Wilson conjecture and in Kazhdan-Lusztig theory for matroids. We study these varieties as…
The Chow polynomial of a matroid is a fundamental invariant whose coefficients exhibit strong positivity properties, including $\gamma$-positivity. We interpret the normalized Chow coefficients as a probability distribution and establish…