The proof follows from relations between Adem relations (4), using what Bousfield calls “pension operators”, i.e. selfmaps of tensor powers which preserve Adem relations. I believe Bousfield had a proof of this sort. Following Mahowald’s suggestion, we’ll give an EHP proof of the basis. Let V be the Z/2 vectorspace with basis {λp : p ≥ −1}. Define e : V → V by e(λp) = λp+1, and define the selfm...

Let J ⊂ S = K[x0, . . . , xn] be a monomial strongly stable ideal. The collection Mf(J) of the homogeneous polynomial ideals I, such that the monomials outside J form a K-vector basis of S/I, is called a J-marked family. It can be endowed with a structure of affine scheme, called a J-marked scheme. For special ideals J , J-marked schemes provide an open cover of the Hilbert scheme Hilbnp(t), wh...

A new O(n) algorithm is given for evaluating univariate polynomials of degree n in the P olya basis. Since the Lagrange, Bernstein, and monomial bases are all special instances of the P olya basis, this technique leads to e cient evaluation algorithms for these special bases. For the monomial basis, this algorithm is shown to be equivalent to Horner's rule. 1 P olya basis functions Let nk(t) be...

We investigate Gröbner bases of contraction ideals under monomial homomorphisms. As an application, we generalize the result of Aoki–Hibi–Ohsugi– Takemura and Ohsugi–Hibi for toric ideals of nested configurations.

We describe the cell structure of the aane Temperley{Lieb algebra with respect to a monomial basis. We construct a diagram calculus for this algebra.

Three division algorithms are presented for univariate Bernstein polynomials: an algorithm for finding the quotient and remainder of two univariate polynomials, an algorithm for calculating the GCD of an arbitrary collection of univariate polynomials, and an algorithm for computing a μ-basis for the syzygy module of an arbitrary collection of univariate polynomials. Division algorithms for mult...

In their classic 1914 paper, Polýa and Schur introduced and characterized two types of linear operators acting diagonally on the monomial basis of R[x], sending real-rooted polynomials (resp. polynomials with all nonzero roots of the same sign) to real-rooted polynomials. Motivated by fundamental properties of amoebae and discriminants discovered by Gelfand, Kapranov, and Zelevinsky, we introdu...

We present a vertex operator algebra which is an extension of the level k vertex operator algebra for the ŝl2 conformal field theory. We construct monomial basis of its irreducible representations.

Let n, k, and r be nonnegative integers and let Sn be the symmetric group. We introduce a quotient Rn,k,r of the polynomial ring Q[x1, . . . , xn] in n variables which carries the structure of a graded Sn-module. When r > n or k = 0 the quotient Rn,k,r reduces to the classical coinvariant algebra Rn attached to the symmetric group. Just as algebraic properties of Rn are controlled by combinator...

We discuss the construction of volume-preserving splitting methods based on a tensor product of single-variable basis functions. The vector field is decomposed as the sum of elementary divergence-free vector fields (EDFVFs), each of them corresponding to a basis function. The theory is a generalization of the monomial basis approach introduced in Xue & Zanna (2013, BIT Numer. Math., 53, 265–281...