Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA proof of the first Kac-Weisfeiler conjecture in large characteristics
68
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra $mathfrak{g}$. The first predicts the maximal dimension of simple $mathfrak{g}$-modules and in this paper we apply the Lefschetz principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of $mathfrak{gl}_n(k)$ whenever $k$ is an algebraically closed field of characteristic $p gg n$. As a consequence we deduce that the conjecture holds for the the Lie algebra of a group scheme when specialised to an algebraically closed field of almost any characteristic. In the appendix to this paper, written by Akaki Tikaradze, a short proof of the first Kac--Weisfeiler conjecture is given for the Lie algebra of group scheme over a finitely generated ring $R subseteq mathbb{C}$, after base change to a field of large positive characteristic.
rate research
Read More
We present a proof of the compositional shuffle conjecture, which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector space $V_*$ whose degree zero part is the ring of symmetric functions $Sym[X]$ over $mathbb{Q}(q,t)$. We then extend these operators to an action of an algebra $tilde{AA}$ acting on this space, and interpret the right generalization of the $ abla$ using an involution of the algebra which is antilinear with respect to the conjugation $(q,t)mapsto (q^{-1},t^{-1})$.
Auslander-Reiten conjecture, which says that an Artin algebra does not have any non-projective generator with vanishing self-extensions in all positive degrees, is shown to be invariant under certain singular equivalences induced by adjoint pairs, which occur often in matrix algebras, recollements and change of rings. Accordingly, several reduction methods are established to study this conjecture.
In this paper we prove the Aubert-Baum-Plymen-Solleveld conjecture for the split classical groups and establish the connection with the Langlands correspondence. To do this, we review the notion of cuspidality for enhanced Langlands parameters and also review the notion of cuspidal support for enhanced Langlands parameters for split classical groups, both introduced by the author in earlier work.
We prove that the `Upper Matching Conjecture of Friedland, Krop, and Markstrom and the analogous conjecture of Kahn for independent sets in regular graphs hold for all large enough graphs as a function of the degree. That is, for every $d$ and every large enough $n$ divisible by $2d$, a union of $n/(2d)$ copies of the complete $d$-regular bipartite graph maximizes the number of independent sets and matchings of size $k$ for each $k$ over all $d$-regular graphs on $n$ vertices. To prove this we utilize the cluster expansion for the canonical ensemble of a statistical physics spin model, and we give some further applications of this method to maximizing and minimizing the number of independent sets and matchings of a given size in regular graphs of a given minimum girth.
A typical decomposition question asks whether the edges of some graph $G$ can be partitioned into disjoint copies of another graph $H$. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with $n$ edges packs $2n+1$ times into the complete graph $K_{2n+1}$. In this paper, we prove this conjecture for large $n$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
