Framed motives of smooth affine pairs

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category $\mathbf{SH}(k)$ in terms of Voevodsky's framed correspondences. In particular, the motivically fibrant Ω-resolution in positive degrees of the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\infty }{X}_{+}$, where $X_{+}=X??$, for a smooth scheme $X\in {\mathrm{Sm}}_k$ over an infinite perfect field k, is computed.The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres $({\mathbb{A}}^l\times X)/(({\mathbb{A}}^l?0)\times X)$, $X\in {\mathrm{Sm}}_k$, is one of ingredients in the theory. In the article we extend this result to the case of a pair $(X,U)$ given by a smooth affine variety X over k and an open subscheme $U?X$.The result gives an explicit motivically fibrant Ω-resolution in positive degrees for the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\infty }(X_{+}/U_{+})$ of the quotient-sheaf $X_{+}/U_{+}$.     

Permutation trinomials over Fq3

We consider four classes of polynomials over the fields ${\mathbb{F}}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+A{x}^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+A{x}^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+A{x}^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+A{x}^q-Bx$, where $A,B\in {\mathbb{F}}_q$. We find sufficient conditions on the pairs $(A,B)$ for which these polynomials permute ${\mathbb{F}}_{q^3}$ and we give lower bounds on the number of such pairs.     

On discrete Brunn-Minkowski and isoperimetric type inequalities

We show that the lattice point enumerator ${\mathrm{G}}_n(?)$ satisfies${\mathrm{G}}_n{(tK+sL+{(?1,?t+s?)}^n)}^{1/n}\ge t{\mathrm{G}}_n{(K)}^{1/n}+s{\mathrm{G}}_n{(L)}^{1/n}$ for any $K,L?{\mathbb{R}}^n$ bounded sets with integer points and all $t,s\ge 0$.We also prove that a certain family of compact sets, extending that of cubes ${[?m,m]}^n$, with $m\in \mathbb{N}$, minimizes the functional ${\mathrm{G}}_n(K+t{[?1,1]}^n)$, for any $t\ge 0$, among those bounded sets $K?{\mathbb{R}}^n$ with given positive lattice point enumerator.Finally, we show that these new discrete inequalities imply the corresponding classical Brunn-Minkowski and isoperimetric inequalities for non-empty compact sets.     

Reflection length with two parameters in the asymptotic representation theory of type B/C and applications

We introduce a two-parameter function ${?}_{q_{+},q_{?}}$ on the infinite hyperoctahedral group, which is a bivariate refinement of the reflection length. We show that this signed reflection function ${?}_{q_{+},q_{?}}$ is positive definite if and only if it is an extreme character of the infinite hyperoctahedral group and we classify the corresponding set of parameters $q_{+},q_{?}$. We construct the corresponding representations through a natural action of the hyperoctahedral group $B(n)$ on the tensor product of n copies of a vector space, which gives a two-parameter analog of the classical construction of Schur–Weyl.We apply our classification to construct a cyclic Fock space of type B generalizing the one-parameter construction in type A found previously by Bo?ejko and Guta. We also construct a new Gaussian operator acting on the cyclic Fock space of type B and we relate its moments with the Askey–Wimp–Kerov distribution by using the notion of cycles on pair-partitions, which we introduce here. Finally, we explain how to solve the analogous problem for the Coxeter groups of type D by using our main result.