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Register a new userSemi-invariants of Binary Forms and Sylvesters Theorem
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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.
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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.
This work lies across three areas (in the title) of investigation that are by themselves of independent interest. A problem that arose in quantum computing led us to a link that tied these areas together. This link consists of a single formal power series with a multifaced interpretation. The deeper exploration of this link yielded results as well as methods for solving some numerical problems in each of these separate areas.
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.
Let $lambda$ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let $mathrm{QLS}(lambda)$ denote the quantum Lakshmibai-Seshadri (QLS) paths of shape $lambda$. For an element $w$ of a finite Weyl group $W$, the specializations at $t = 0$ and $t = infty$ of the nonsymmetric Macdonald polynomial $E_{w lambda}(q, t)$ are explicitly described in terms of QLS paths of shape $lambda$ and the degree function defined on them. Also, for (level-zero) dominant integral weights $lambda$, $mu$, we have an isomorphism $Theta : mathrm{QLS}(lambda + mu) rightarrow mathrm{QLS}(lambda) otimes mathrm{QLS}(mu)$ of crystals. In this paper, we study the behavior of the degree function under the isomorphism $Theta$ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules.
For a given sequence $mathbf{alpha} = [alpha_1,alpha_2,dots,alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(mathbf{alpha})(t)$ that counts the nonnegative integer solutions of the equation $alpha_1x_1+alpha_2 x_2+cdots+alpha_{N} x_{N}+alpha_{N+1}x_{N+1}=t$, where the right-hand side $t$ is a varying nonnegative integer. It is well-known that $E(mathbf{alpha})(t)$ is a quasi-polynomial function in the variable $t$ of degree $N$. In combinatorial number theory this function is known as Sylvesters denumerant. Our main result is a new algorithm that, for every fixed number $k$, computes in polynomial time the highest $k+1$ coefficients of the quasi-polynomial $E(mathbf{alpha})(t)$ as step polynomials of $t$ (a simpler and more explicit representation). Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for $E(mathbf{alpha})(t)$ and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Our algorithm also uses Barvinoks fundamental fast decomposition of a polyhedral cone into unimodular cones. This paper also presents a simple algorithm to predict the first non-constant coefficient and concludes with a report of several computational experiments using an implementation of our algorithm in LattE integrale. We compare it with various Maple programs for partial or full computation of the denumerant.
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