In this paper we show that there is a link between the combinatorics of the canonical basis of a quantized enveloping algebra and the monomial bases of the second author [21] arising from representations of quivers. We prove that some reparametrization functions of the canonical basis, arising from the link between Lusztig’s approach to the canonical basis and the string parametrization of the ...

We give matrices of which their determinants are the Jack polynomials expanded in terms of the monomial basis. The top row of this matrix is a list of monomial functions, the entries of the sub-diagonal are of the form −(rα + s), with r and s ∈ +, the entries above the sub-diagonal are nonnegative integers, and below all entries are 0. The quasi-triangular nature of this matrix gives a recursio...

Classically, Gröbner bases are computed by first prescribing a set monomial order. Moss Sweedler suggested an alternative and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on k(x, y) that are suitable for this framework. For these valuations, we compute ν(k[x, y]∗) and use this ...

In the classical paper [PS14] I. Schur and G. Polýa introduced and characterized two types of multiplier sequences, i.e. linear operators acting diagonally in the monomial basis of R[x] and sending real-rooted (resp. with all real roots of the same sign) polynomials to real-rooted polynomials. Motivated by a fundamental property of amoebae and discriminants discovered in [GKZ94] we introduce be...

Let A be a subvariety of affine space A whose irreducible components are d-dimensional linear or affine subspaces of A. Denote by D(A) ⊂ N the set of exponents of standard monomials ofA. Using the Hilbert function, we show thatD(A) contains as many subspaces of dimension d as A contains irreducible components. We refine this result in various ways. Firstly, we specify the directions into which ...

Let A be a finite set of closed rational points in projective space, let I be the vanishing ideal of A , and let D(A ) be the set of exponents of those monomials which do not occur as leading monomials of elements of I . We show that the size of A equals the number of axes contained in D(A ). Furthermore, we present an algorithm for the construction of the Gröbner basis of I (A ), hence also of...

‎Let $mathcal{A}_p$ be the mod $p$ Steenrod algebra‎, ‎where $p$ is an odd prime‎, ‎and let $mathcal{A}$ be the‎ subalgebra $mathcal{A}$ of $mathcal{A}_p$ generated by the Steenrod $p$th powers‎. ‎We generalize the $X$-basis in $mathcal{A}$ to $mathcal{A}_p$‎.

Let A be a finite set of closed rational points in projective space, let I be the vanishing ideal of A , and let D(A ) be the set of exponents of those monomials which do not occur as leading monomials of elements of I . We show that the size of A equals the number of axes contained in D(A ). Furthermore, we present an algorithm for the construction of the Gröbner basis of I (A ), hence also of...