Let $\Omega _n$ be the ring of polynomial-valued holomorphic differential forms on complex n-space, referred to in physics as the superspace ring of rank n. The symmetric group ${\mathfrak {S}}_n$ acts diagonally on $\Omega _n$ by permuting commuting and anticommuting generators simultaneously. We let $SI_n \subseteq \Omega _n$ be the ideal generated by ${\mathfrak {S}}_n$-invariants with vanishing constant term and study the quotient $SR_n = \Omega _n / SI_n$ of superspace by this ideal. We calculate the doubly-graded Hilbert series of $SR_n$ and prove an ‘operator theorem’, which characterizes the harmonic space $SH_n \subseteq \Omega _n$ attached to $SR_n$ in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results that were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Colmenarejo, Li, Machacek, Sulzgruber, Swanson, Wallach and Zabrocki.

Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are strong Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of surfaces with orbifold points, establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their singularity categories, their Gorenstein dimensions and derived invariants such as the determinant of their q-Cartan matrices.

Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

Frieze patterns, as introduced by Coxeter in the 1970s, are closely related to cluster algebras without coefficients. A suitable generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp. In this paper, we study these frieze patterns with coefficients systematically and prove various fundamental results, generalizing classic results for frieze patterns. As a consequence, we see how frieze patterns with coefficients can be obtained from classic frieze patterns by cutting out subpolygons from the triangulated polygons associated with classic Conway–Coxeter frieze patterns. We address the question of which frieze patterns with coefficients can be obtained in this way and solve this problem completely for triangles. Finally, we prove a finiteness result for frieze patterns with coefficients by showing that for a given boundary sequence there are only finitely many (nonzero) frieze patterns with coefficients with entries in a subset of the complex numbers without an accumulation point.

We study the equivariant oriented cohomology ring $\mathtt{h}_{T}(G/P)$ of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott–Samelson classes in $\mathtt{h}_{T}(G/P)$ can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results of Chow groups by Brion, Knutson, Peterson, Tymoczko and others. Our main result concerns the equivariant oriented cohomology theory $\mathfrak{h}$ corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar’s parabolic Kazhdan–Lusztig basis, i.e., we realize it as some cohomology classes (the parabolic Kazhdan–Lusztig (KL) Schubert classes) in $\mathfrak{h}_{T}(G/P)$. We make a positivity conjecture, and a conjecture about the relationship of such classes with smoothness of Schubert varieties. We also prove the latter in several special cases.

The set of row reduced matrices (and of echelon form matrices) is closed under multiplication. We show that any system of representatives for the $\text{Gl}_{n}(\mathbb{K})$ action on the $n\times n$ matrices, which is closed under multiplication, is necessarily conjugate to one that is in simultaneous echelon form. We call such closed representative systems Grassmannian semigroups. We study internal properties of such Grassmannian semigroups and show that they are algebraic semigroups and admit gradings by the finite semigroup of partial order preserving permutations, with components that are naturally in one–one correspondence with the Schubert cells of the total Grassmannian. We show that there are infinitely many isomorphism types of such semigroups in general, and two such semigroups are isomorphic exactly when they are semiconjugate in $M_{n}(\mathbb{K})$. We also investigate their representation theory over an arbitrary field, and other connections with multiplicative structures on Grassmannians and Young diagrams.

We present a stand-alone simple proof of a probabilistic interpretation of the Gaussian binomial coefficients by conditioning a random walk to hit a given lattice point at a given time.

We consider Schubert problems with respect to flags osculating the rational normal curve. These problems are of special interest when the osculation points are all real. In this case, for zero-dimensional Schubert problems, the solutions are “ as real as possible”. Recent work by Speyer has extended the theory to the moduli space $\overline{{{\mathcal{M}}_{0,\,r}}}$ allowing the points to collide. This gives rise to smooth covers $\overline{{{\mathcal{M}}_{0,\,r}}}\left( \mathbb{R} \right)$, with structure and monodromy described by Young tableaux and jeu de taquin.

In this paper, we give analogous results on one-dimensional Schubert problems over $\overline{{{\mathcal{M}}_{0,\,r}}}$.Their(real) geometry turns out to be described by orbits of Schützenberger promotion and a related operation involving tableau evacuation. Over ${{\mathcal{M}}_{0,\,r}}$, our results show that the real points of the solution curves are smooth.

We also find a new identity involving “first-order” $\text{K}$-theoretic Littlewood-Richardson coefficients, for which there does not appear to be a known combinatorial proof.

This paper concerns a relatively new combinatorial structure called staircase tableaux. They were introduced in the context of the asymmetric exclusion process and Askey–Wilson polynomials; however, their purely combinatorial properties have gained considerable interest in the past few years.

In this paper we further study combinatorial properties of staircase tableaux. We consider a general model of random staircase tableaux in which symbols (Greek letters) that appear in staircase tableaux may have arbitrary positive weights. (We consider only the case with the parameters u = q = 1.) Under this general model we derive a number of results. Some of our results concern the limiting laws for the number of appearances of symbols in a random staircase tableaux. They generalize and subsume earlier results that were obtained for specific values of the weights.

One advantage of our generality is that we may let the weights approach extreme values of zero or infinity, which covers further special cases appearing earlier in the literature. Furthermore, our generality allows us to analyse the structure of random staircase tableaux, and we obtain several results in this direction.

One of the tools we use is the generating functions of the parameters of interest. This leads us to a two-parameter family of polynomials, generalizing the classical Eulerian polynomials.

We also briefly discuss the relation of staircase tableaux to the asymmetric exclusion process, to other recently introduced types of tableaux, and to an urn model studied by a number of researchers, including Philippe Flajolet.

Let S = K[x1,x2,…,xn] be a polynomial ring over a field K. Let Δ be a simplicial complex whose vertex set is contained in {1, 2,…,n}. For an integer k ≥ 0, we investigate the k-Buchsbaum property of residue class rings S/I(t); and S/It for the Stanley-Reisner ideal I = IΔ. We characterize the k-Buchsbaumness of such rings in terms of the simplicial complex Δ and the power t. We also give a characterization in the case where I is the edge ideal of a simple graph.

Following Radford's proof of Lagrange's theorem for pointed Hopf algebras, we prove Lagrange‘s theorem for Hopf monoids in the category of connected species. As a corollary, we obtain necessary conditions for a given subspecies $\mathbf{k}$ of a Hopf monoid $\mathbf{h}$ to be a Hopf submonoid: the quotient of any one of the generating series of $\mathbf{h}$ by the corresponding generating series of $\mathbf{k}$ must have nonnegative coefficients. Other corollaries include a necessary condition for a sequence of nonnegative integers to be the dimension sequence of a Hopf monoid in the form of certain polynomial inequalities and of a set-theoretic Hopf monoid in the form of certain linear inequalities. The latter express that the binomial transform of the sequence must be nonnegative.

In 1960, Sobolev proved that for a finite reflection group $G$, a $G$-invariant cubature formula is of degree $t$ if and only if it is exact for all $G$-invariant polynomials of degree at most $t$. In this paper, we make some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and, moreover, gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007), which classifies tight Euclidean designs invariant under the Weyl group of type $B$, to other finite reflection groups.

We prove a conjecture of Kontsevich, which asserts that the iterations of the non-commutative rational map Fr:(x,y)→(xyx−1,(1+yr)x−1) are given by non-commutative Laurent polynomials with non-negative integer coefficients.

We consider a class of strongly $q$-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly $q$-log-convex. We also prove that the Bessel transformation preserves log-convexity.

Given a nonnegative integer $m$ and a finite collection $A$ of linear forms on ${{\mathbb{Q}}^{d}}$, the arrangement of affine hyperplanes in ${{\mathbb{Q}}^{d}}$ defined by the equations $\alpha \left( x \right)\,=\,k$ for $\alpha \,\in \,A$ and integers $k\,\in \,\left[ -m,\,m \right]$ is denoted by ${{A}^{m}}$. It is proved that the coefficients of the characteristic polynomial of ${{A}^{m}}$ are quasi-polynomials in $m$ and that they satisfy a simple combinatorial reciprocity law.

Multiplicity-free algebraic geometry is the study of subvarieties $Y\,\subseteq \,X$ with the “smallest invariants” as witnessed by a multiplicity-free Chow ring decomposition of $\left[ Y \right]\,\in \,{{A}^{*}}\left( X \right)$ into a predetermined linear basis.

This paper concerns the case of Richardson subvarieties of the Grassmannian in terms of the Schubert basis. We give a nonrecursive combinatorial classification of multiplicity-free Richardson varieties, i.e., we classify multiplicity-free products of Schubert classes. This answers a question of W. Fulton.

Generalizing the concepts of Stanley-Reisner and affine monoid algebras, one can associate to a rational pointed fan Σ in ℝd the ℤd-graded toric face ring K[Σ]. Assuming that K[Σ] is Cohen-Macaulay, the main result of this paper is to characterize the situation when its canonical module is isomorphic to a ℤd-graded ideal of K[Σ]. From this result several algebraic and combinatorial consequences are deduced. As an application, we give a relation between the cleanness of K[Σ] and the shellability of Σ.

Let $X$ be a minuscule Schubert variety. In this paper, we associate a quiver with $X$ and use the combinatorics of this quiver to describe all relative minimal models $\widehat{\pi}:{\widehat{X}}\to X$. We prove that all the morphisms $\widehat{\pi}$ are small and give a combinatorial criterion for $\widehat{X}$ to be smooth and thus a small resolution of $X$. We describe in this way all small resolutions of $X$. As another application of this description of relative minimal models, we obtain the following more intrinsic statement of the main result of Perrin, J. Algebra 294 (2005), 431–462. Let $\alpha\in A_1(X)$ be an effective 1-cycle class. Then the irreducible components of the scheme Hom$_{\alpha}(p^1,X)$ of morphisms from $\mathbb{P}^1$ to $X$ and of class $\alpha$ are indexed by the set: ${\mathfrak{ne}}(\alpha)=\{\beta\in A_1(\widehat{X}) \mid \beta$ is effective and $\widehat{\pi}_*\beta=\alpha\}$ which is independent of the choice of a relative minimal model $\widehat{X}$ of $X$.

We count the number of strictly positive $B$-stable ideals in the nilradical of a Borel subalgebra and prove that the minimal roots of any $B$-stable ideal are conjugate by an element of the Weyl group to a subset of the simple roots. We also count the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.