Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSome implications of a conjecture of Zabrocki to the action of $S_{n}$ on polynomial differential forms
57
0
0.0
(
0
)
Authors
Nolan R. Wallach
Ask ChatGPT about the research
No Arabic abstract
The symmetric group acts on polynomial differential forms on $mathbb{R}^{n}$ through its action by permuting the coordinates. In this paper the $S_{n}% $-invariants are shown to be freely generated by the elementary symmetric polynomials and their exterior derivatives. A basis of the alternants in the quotient of the ideal generated by the homogeneous invariants of positive degree is given. In addition, the highest bigraded degrees are given for the quotient. All of these results are consistent with predictions derived by Garsia and Romero from a recent conjecture of Zabrocki.
rate research
Read More
The symmetric difference of the $q$-binomial coefficients $F_{n,k}(q)={n+kbrack k}-q^{n}{n+k-2brack k-2}$ was introduced by Reiner and Stanton. They proved that $F_{n,k}(q)$ is symmetric and unimodal for $k geq 2$ and $n$ even by using the representation theory for Lie algebras. Based on Sylvesters proof of the unimodality of the Gaussian coefficients, as conjectured by Cayley, we find an interpretation of the unimodality of $F_{n,k}(q)$ in terms of semi-invariants. In the spirit of the strict unimodality of the Gaussian coefficients due to Pak and Panova, we prove the strict unimodality of the symmetric difference $G_{n,k,r}(q)={n+kbrack k}-q^{nr/2}{n+k-rbrack k-r}$, except for the two terms at both ends, where $n,rgeq8$, $kgeq r$ and at least one of $n$ and $r$ is even.
We obtain a combinatorial formula related to the shear transformation for semi-invariants of binary forms, which implies the classical characterization of semi-invariants in terms of a differential operator. Then, we present a combinatorial proof of an identity of Hilbert, which leads to a relation of Cayley on semi-invariants. This identity plays a crucial role in the original proof of Sylvesters theorem on semi-invariants in connection with the Gaussian coefficients. Moreover, we show that the additivity lemma of Pak and Panova which yields the strict unimodality of the Gaussian coefficients for $n,k geq 8$ can be deduced from the ring property of semi-invariants.
Let $W$ be a finite irreducible real reflection group, which is a Coxeter group. We explicitly construct a basis for the module of differential 1-forms with logarithmic poles along the Coxeter arrangement by using a primitive derivation. As a consequence, we extend the Hodge filtration, indexed by nonnegative integers, into a filtration indexed by all integers. This filtration coincides with the filtration by the order of poles. The results are translated into the derivation case.
The Hall ratio of a graph $G$ is the maximum value of $v(H) / alpha(H)$ taken over all non-null subgraphs $H$ of $G$. For any graph, the Hall ratio is a lower-bound on its fractional chromatic number. In this note, we present various constructions of graphs whose fractional chromatic number grows much faster than their Hall ratio. This refutes a conjecture of Harris.
In the 90s a collection of Plethystic operators were introduced in [3], [7] and [8] to solve some Representation Theoretical problems arising from the Theory of Macdonald polynomials. This collection was enriched in the research that led to the results which appeared in [5], [6] and [9]. However since some of the identities resulting from these efforts were eventually not needed, this additional work remained unpublished. As a consequence of very recent publications [4], [11], [19], [20], [21], a truly remarkable expansion of this theory has taken place. However most of this work has appeared in a language that is virtually inaccessible to practitioners of Algebraic Combinatorics. Yet, these developments have led to a variety of new conjectures in [2] in the Combinatorics and Symmetric function Theory of Macdonald Polynomials. The present work results from an effort to obtain in an elementary and accessible manner all the background necessary to construct the symmetric function side of some of these new conjectures. It turns out that the above mentioned unpublished results provide precisely the tools needed to carry out this project to its completion.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
