Edward E. Allen

The Descent Monomials and a Basis for the Diagonally Symmetric Polynomials

AbstractLet R(X) = Q[x1, x2, ..., xn] be the ring of polynomials in the variables X = {x1, x2, ..., xn} and R*(X) denote the quotient of R(X) by the ideal generated by the elementary symmetric functions. Given a σ ∈ Sn, we let g

Heuristics for dependency conjectures in proteomic signaling pathways

_\sigma (X) = \prod\nolimits_{\sigma _i \succ \sigma _{i + 1} } {(x_{\sigma _1 } x_{\sigma _2 } \ldots x_{\sigma _i } } )

Some Graded Representations of the Complex Reflection Groups

In the late 1970s I. Gessel conjectured that these monomials, called the descent monomials, are a basis for R*(X). Actually, this result was known to Steinberg [10]. A. Garsia showed how it could be derived from the theory of Stanley-Reisner Rings [3]. Now let R(X, Y) denote the ring of polynomials in the variables X = {x1, x2, ..., xn} and Y = {y1, y2, ..., yn}. The diagonal action of σ ∈ Sn on polynomial P(X, Y) is defined as

Reconstructing networks using co-temporal functions

\sigma P(X,Y) = P(x_{\sigma _1 } ,x_{\sigma _2 } , \ldots ,x_{\sigma _n } ,y_{\sigma _1 } ,y_{\sigma _2 } , \ldots ,y_{\sigma _n } )

Comparison of Co-temporal Modeling Algorithms on Sparse Experimental Time Series Data Sets

Let Rρ(X, Y) be the subring of R(X, Y) which is invariant under the diagonal action. Let Rρ*(X, Y) denote the quotient of Rρ(X, Y) by the ideal generated by the elementary symmetric functions in X and the elementary symmetric functions in Y. Recently, A. Garsia in [4] and V. Reiner in [8] showed that a collection of polynomials closely related to the descent monomials are a basis for Rρ*(X, Y). In this paper, the author gives elementary proofs of both theorems by constructing algorithms that show how to expand elements of R*(X) and Rρ*(X, Y) in terms of their respective bases.

Computational modeling of the effects of oxidative stress on the IGF-1 signaling pathway in human articular chondrocytes

Abstract Let μ = (μ1 ⩾ μ2 ⩾ ⋯ ⩾ μk + 1) = (k, 1n − k) be a partition of n. In [GH] Garsia and Haiman show that the diagonal action of Sn on the space of harmonic polynomials H μ affords the left regular representation p of Sn. Furthermore, Garsia and Haiman define a bigraded character of the diagonal action of Sn on H μ and show that the character multiplicities are polynomials K λ, μ (q, t) that are closely related to the Macdonald-Kostka polynomials Kλ, μ(q, t). In this paper we construct a collection of polynomials B(μ) that form a basis for H μ which exhibits the decomposition of H μ into its irreducible parts. Through this connection we give a combinatorial interpretation of the polynomials K λ, μ (q, t) .