The BKR inequality conjectured by van den Berg and Kesten in [11], and proved by Reimer in [8], states that for A and B events on S, a finite product of finite sets Si, i = 1, . . . , n, and P any product measure on S, P (A B) ≤ P (A)P (B), where the set A B consists of the elementary events which lie in both A and B for ‘disjoint reasons.’ Precisely, with n := {1, . . . , n} and K ⊂ n, for x ∈...

We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case where G= SLn, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of sym...

We show how to compute a certain group H l (G) of equivalence classes of invariant Drinfeld twists on the algebra of a finite group G over a field k of characteristic zero. This group is naturally isomorphic to the second lazy cohomology group H l (Ok(G)) of the Hopf algebra Ok(G) of k-valued functions on G. When k is algebraically closed, the answer involves the group of outer automorphisms of...

The paper deals with group dualities. A duality is simply a pair (G,H) where G an abstract abelian and H subgroup of characters defined on G. topology τ compatible the (also called dual pair) if equipped has H. topological (G,τ) gives rise to natural (G,G∧), G∧ stands for continuous We prove that existence g-barrelled (G,G∧) equivalent semireflexivity in Pontryagin's sense endowed pointwise con...

Let G be a finite group and let k be a field of characteristic p. It is known that a kG-module V carries a non-degenerate G-invariant bilinear form b if and only if V is self-dual. We show that whenever a Morita bimodule M which induces an equivalence between two blocks B(kG) and B(kH) of group algebras kG and kH is self-dual then the correspondence preserves self-duality. Even more, if the bil...

let $g$ be a simple graph‎, ‎and $g^{sigma}$‎ ‎be an oriented graph of $g$ with the orientation ‎$sigma$ and skew-adjacency matrix $s(g^{sigma})$‎. ‎the $k-$th skew spectral‎ ‎moment of $g^{sigma}$‎, ‎denoted by‎ ‎$t_k(g^{sigma})$‎, ‎is defined as $sum_{i=1}^{n}( ‎‎‎lambda_{i})^{k}$‎, ‎where $lambda_{1}‎, ‎lambda_{2},cdots‎, ‎lambda_{n}$ are the eigenvalues of $g^{sigma}$‎. ‎suppose‎ ‎$g^{sigma...

A set P of disjoint paths in a graph G is called a (complete) path cover of G if every vertex of G belongs to one of the paths in P. A path cover of any subgraph of G is called a partial path cover of G. For xed k > 0, a k-blanket of graph G is a partial path cover P of G, consisting of exactly k paths, that maximizes the size of the subgraph covered by P. A k-core of graph G is a partial path ...

Let μ denote the Gaussian measure on n×k defined by dμ (Z) = −nk exp −Tr ZZ dZ, where Tr denotes the trace function, Z = Z̄ , and dZ denotes the Lebesgue measure on n×k . Let Fn×k denote the Bargmann–Segal–Fock space of holomorphic entire functions on n×k which are also square-integrable with respect to μ. Fix n and let Fn denote the Hilbert-space completion of the inductive limit limk Fn×k . Le...