Related papers: Planar Shuffle Product, Co-Addition and the non-as…
The expansion of a Schubert polynomial into slide polynomials corresponds to a sum over sub-balls in the subword complex. There has been recent interest in other, coarser, expansions of Schubert polynomials. We extend the methods used in…
We give new product formulas for the number of standard Young tableaux of certain skew shapes and for the principal evaluation of the certain Schubert polynomials. These are proved by utilizing symmetries for evaluations of factorial Schur…
We define slope subalgebras in the shuffle algebra associated to a (doubled) quiver, thus yielding a factorization of the universal R-matrix of the double of the shuffle algebra in question. We conjecture that this factorization matches the…
We study scalar products of Bethe vectors in the models solvable by the nested algebraic Bethe ansatz and described by $\mathfrak{gl}(m|n)$ superalgebra. Using coproduct properties of the Bethe vectors we obtain a sum formula for their…
In this paper, we are going to perform the shuffle products of $Z_-(n) = \sum_{a+b=m} (-1)^{b} \zeta(\{1\}^{a},b+2)$ and $Z_+^\star(n) = \sum_{c+d=n} \zeta^{\star}(\{1\}^{c},d+2)$ with $m+n = p$. The resulted shuffle relation is a weighted…
We prove Stanley's conjecture that, if delta_n is the staircase shape, then the skew Schur functions s_{delta_n / mu} are non-negative sums of Schur P-functions. We prove that the coefficients in this sum count certain fillings of shifted…
The shuffle algebra on positive integers encodes the usual multiple zeta values (MZVs) (with positive arguments) thanks to the representations of MZVs by iterated Chen integrals of Kontsevich. Together with the quasi-shuffle (stuffle)…
Multiple Dedekind zeta values were recently defined by the second author. In a separate paper, the second author constructed double shuffle relations in some cases as a response to questions asked by Richard Hain and Alexander Goncharov. In…
We introduce the shuffle of deformed permutahedra (a.k.a. generalized permutahedra), a simple associative operation obtained as the Cartesian product followed by the Minkowski sum with the graphical zonotope of a complete bipartite graph.…
In order to extend the Sch\"utzenberger's factorization to general perturbations, the combinatorial aspects of the Hopf algebra of the $\phi$-deformed stuffle product is developed systematically in a parallel way with those of the shuffle…
Consider a nonzero contraction $T$ and a bounded operator $X$ satisfying $TX=qXT$ for a complex number $q$. There are some interesting results in the literature on $q$-commuting dilation and $q$-commutant lifting of such pair $(T,X)$ when…
In this paper the analogy between differential forms arising from integrals in additive calculus and forms arising from the integrals in product calculus is investigated. It is found that with an appropriate definition of scalar…
We discuss a product formula for $F$-polynomials in cluster algebras, and provide two proofs. One proof is inductive and uses only the mutation rule for $F$-polynomials. The other is based on the Fock-Goncharov decomposition of mutations.…
For any lattice polytope $P$, we consider an associated polynomial $\bar{\delta}_{P}(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known…
A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the…
We define an elliptic generating series whose coefficients, the elliptic multizetas, are related to the elliptic analogues of multiple zeta values introduced by Enriquez as the coefficients of his elliptic associator; both sets of…
The operator valued distributions which arise in quantum field theory on the noncommutative Minkowski space can be symbolized by a generalization of chord diagrams, the dotted chord diagrams. In this framework, the combinatorial aspects of…
We equip the graded polynomial algebra generated by nonplanar rooted binary trees with a Hopf algebra structure by defining a coproduct which disallows cutting both children of any given vertex, refining Connes-Kreimer's notion of…
The commutative trigonometric shuffle algebra ${\mathrm A}$ is a space of symmetric rational functions satisfying certain wheel conditions. We describe a ring isomorphism between ${\mathrm A}$ and the center of the Hecke algebra using a…
We define and study the shuffle algebra $Sh_{m|n}$ of the quantum toroidal algebra $\mathcal E_{m|n}$ associated to Lie superalgebra $\mathfrak{gl}_{m|n}$. We show that $Sh_{m|n}$ contains a family of commutative subalgebras $\mathcal…