Vasu Tewari
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View article: The quasisymmetric flag variety: a toric complex on noncrossing partitions
The quasisymmetric flag variety: a toric complex on noncrossing partitions Open
We develop the geometric theory of equivariant quasisymmetry via a new ``quasisymmetric flag variety''. This is a toric complex in the flag variety whose fixed point set is the set of noncrossing partitions, and whose cohomology ring is th…
View article: Equivariant quasisymmetry and noncrossing partitions
Equivariant quasisymmetry and noncrossing partitions Open
We introduce a definition of ``equivariant quasisymmetry'' for polynomials in two sets of variables. Using this definition we define quasisymmetric generalizations of the theory of double Schur and double Schubert polynomials that we call …
View article: Schubert polynomial expansions revisited
Schubert polynomial expansions revisited Open
We give an elementary approach utilizing only the divided difference formalism for obtaining expansions of Schubert polynomials that are manifestly nonnegative, by studying solutions to the equation $\sum Y_i\partial _i=\operatorname {id}$…
View article: The geometry of quasisymmetric coinvariants
The geometry of quasisymmetric coinvariants Open
We develop a quasisymmetric analogue of the theory of Schubert cycles, building off of our previous work on a quasisymmetric analogue of Schubert polynomials and divided differences. Our constructions result in a natural geometric interpre…
View article: \(P\)-partitions with flags and back stable quasisymmetric functions
\(P\)-partitions with flags and back stable quasisymmetric functions Open
Stanley's theory of \\((P,\\omega)\\)-partitions is a standard tool in combinatorics. It can be extended to allow for the presence of a restriction, that is a given maximal value for partitions at each vertex of the poset, as was shown by …
View article: Quasisymmetric divided differences
Quasisymmetric divided differences Open
We develop a quasisymmetric analogue of the combinatorial theory of Schubert polynomials and the associated divided difference operators. Our counterparts are "forest polynomials", and a new family of linear operators, whose theory of comp…
View article: Tutte polynomials in superspace
Tutte polynomials in superspace Open
We associate a quotient of superspace to any hyperplane arrangement by considering the differential closure of an ideal generated by powers of certain homogeneous linear forms. This quotient is a superspace analogue of the external zonotop…
View article: $[\text{Perm}_n]$ via $\text{QSym}_n^+$
$[\text{Perm}_n]$ via $\text{QSym}_n^+$ Open
International audience
View article: Forest polynomials and the class of the permutahedral variety
Forest polynomials and the class of the permutahedral variety Open
We study a basis of the polynomial ring that we call forest polynomials. This family of polynomials is indexed by a combinatorial structure called indexed forests and permits several definitions, one of which involves flagged P-partitions.…
View article: $P$-partitions with flags and back stable quasisymmetric functions
$P$-partitions with flags and back stable quasisymmetric functions Open
Stanley's theory of $(P,ω)$-partitions is a standard tool in combinatorics. It can be extended to allow for the presence of a restriction, that is a given maximal value for partitions at each vertex of the poset, as was shown by Assaf and …
View article: Remixed Eulerian numbers
Remixed Eulerian numbers Open
Remixed Eulerian numbers are a polynomial q -deformation of Postnikov’s mixed Eulerian numbers. They arose naturally in previous work by the authors concerning the permutahedral variety and subsume well-known families of polynomials such a…
View article: Down-up algebras and chromatic symmetric functions
Down-up algebras and chromatic symmetric functions Open
We establish Guay-Paquet's unpublished linear relation between certain chromatic symmetric functions by relating his algebra on paths to the $q$-Klyachko algebra. The coefficients in this relations are $q$-hit polynomials, and they come up…
View article: Remixed Eulerian numbers
Remixed Eulerian numbers Open
Remixed Eulerian numbers are a polynomial $q$-deformation of Postnikov's mixed Eulerian numbers. They arose naturally in previous work by the authors concerning the permutahedral variety and subsume well-known families of polynomials such …
View article: Zonotopal algebras, orbit harmonics, and Donaldson-Thomas invariants of symmetric quivers
Zonotopal algebras, orbit harmonics, and Donaldson-Thomas invariants of symmetric quivers Open
We apply the method of orbit harmonics to the set of break divisors and orientable divisors on graphs to obtain the central and external zonotopal algebras respectively. We then relate a construction of Efimov in the context of cohomologic…
View article: Divisors on complete multigraphs and Donaldson-Thomas invariants of loop quivers
Divisors on complete multigraphs and Donaldson-Thomas invariants of loop quivers Open
We study the action of $S_n$ on the set of break divisors on complete multigraphs $K_{n}^m$. We provide an alternative characterization for these divisors, by virtue of which we show that orbits of this action are enumerated by the numeric…
View article: Divisors on complete multigraphs and Donaldson-Thomas invariants of loop\n quivers
Divisors on complete multigraphs and Donaldson-Thomas invariants of loop\n quivers Open
We study the action of $S_n$ on the set of break divisors on complete\nmultigraphs $K_{n}^m$. We provide an alternative characterization for these\ndivisors, by virtue of which we show that orbits of this action are enumerated\nby the nume…
View article: The Permutahedral Variety, Mixed Eulerian Numbers, and Principal Specializations of Schubert Polynomials
The Permutahedral Variety, Mixed Eulerian Numbers, and Principal Specializations of Schubert Polynomials Open
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 normalized mixed Eulerian numbers indexed naturally by reduce…
View article: Trimming the permutahedron to extend the parking space
Trimming the permutahedron to extend the parking space Open
Berget and Rhoades asked whether the permutation representation obtained by the action of $S_{n-1}$ on parking functions of length $n-1$ can be extended to a permutation action of $S_{n}$. We answer this question in the affirmative. We rea…
View article: A $q$-deformation of an algebra of Klyachko and Macdonald's reduced word formula
A $q$-deformation of an algebra of Klyachko and Macdonald's reduced word formula Open
There is a striking similarity between Macdonald's reduced word formula and the image of the Schubert class in the cohomology ring of the permutahedral variety $\mathrm{Perm}_n$ as computed by Klyachko. Toward understanding this better, we…
View article: Noncommutative LR coefficients and crystal reflection operators
Noncommutative LR coefficients and crystal reflection operators Open
We relate noncommutative Littlewood–Richardson coefficients of Bessenrodt–Luoto–van Willigenburg to classical Littlewood–Richardson coefficients via crystal reflection operators. A key role is played by the combinatorics of frank words.
View article: Alternating sign matrices and totally symmetric plane partitions
Alternating sign matrices and totally symmetric plane partitions Open
We study the Schur polynomial expansion of a family of symmetric polynomials related to the refined enumeration of alternating sign matrices with respect to their inversion number, complementary inversion number and the position of the uni…
View article: The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials
The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials Open
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…
View article: Divided symmetrization and quasisymmetric functions (extended abstract)
Divided symmetrization and quasisymmetric functions (extended abstract) Open
International audience
View article: A noncommutative geometric LR rule
A noncommutative geometric LR rule Open
The geometric Littlewood-Richardson (LR) rule is a combinatorial algorithm for computing LR coefficients derived from degenerating the Richardson variety into a union of Schubert varieties in the Grassmannian. Such rules were first given b…
View article: Boolean Product Polynomials, Schur Positivity, and Chern Plethysm
Boolean Product Polynomials, Schur Positivity, and Chern Plethysm Open
Let $k \leq n$ be positive integers, and let $X_n = (x_1, \dots , x_n)$ be a list of $n$ variables. The Boolean product polynomial $B_{n,k}(X_n)$ is the product of the linear forms $\sum _{i \in S} x_i$, where $S$ ranges over all $k$-eleme…
View article: Chromatic nonsymmetric polynomials of Dyck graphs are slide-positive
Chromatic nonsymmetric polynomials of Dyck graphs are slide-positive Open
Motivated by the study of Macdonald polynomials, J. Haglund and A. Wilson introduced a nonsymmetric polynomial analogue of the chromatic quasisymmetric function called the \emph{chromatic nonsymmetric polynomial} of a Dyck graph. We give a…