We define an operator α on ℂ³ ⊗ ℂ³ associated with the quantum group $U_q\left(2\right)$, which satisfies the Yang-Baxter equation and a cubic equation (α² - 1)(α + q²) = 0. This operator can be extended to a family of operators $h_j:=I_j\otimes \alpha \otimes I_{n-2-j}$ on ${\left(ℂ³\right)}^{\otimes n}$ with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra ${\mathscr{H}}_{q,n}\left(2\right)$ associated with the quantum group $U_q\left(2\right)$. The purpose of this note is to present the construction.

Pólya’s fundamental enumeration theorem and some results from Williamson’s generalized setup of it are proved in terms of Schur- Macdonald’s theory (S-MT) of “invariant matrices”. Given a permutation group W ≤ Sd and a one-dimensional character χ of W , the polynomial functor Fχ corresponding via S-MT to the induced monomial representation Uχ = ind|Sdv/W (χ) of Sd , is studied. It turns out that the characteristic ch(Fχ ) is the weighted inventory of some set J(χ) of W -orbits in the integer-valued hypercube...

In this paper, the unproven half of Richard Brauer’s Height Zero Conjecture is reduced to a question on simple groups.

We study equivalences for category ${𝒪}_p$ of the rational Cherednik algebras ${𝐇}_p$ of type $G_{\ell }\left(n\right)={\left({\mu }_{\ell }\right)}^n⋊{𝔖}_n$: a highest weight equivalence between ${𝒪}_p$ and ${𝒪}_{\sigma \left(p\right)}$ for $\sigma \in {𝔖}_{\ell }$ and an action of ${𝔖}_{\ell }$ on an explicit non-empty Zariski open set of parameters $p$; a derived equivalence between ${𝒪}_p$ and ${𝒪}_{p^{\text{'}}}$ whenever $p$ and $p^{\text{'}}$ have integral difference; a highest weight equivalence between ${𝒪}_p$ and a parabolic category $𝒪$ for the general linear group, under a non-rationality assumption on the parameter $p$. As a consequence, we confirm special cases of conjectures...

The $n$-fold product $X^n$ of an arbitrary space usually supports only the obvious permutation action of the symmetric group ${\Sigma }_n$. However, if $X$ is a $p$-complete, homotopy associative, homotopy commutative $H$-space one can define a homotopy action of ${\mathrm{GL}}_n\left({\mathbb{Z}}_p\right)$ on $X^n$. In various cases, e.g. if multiplication by $p^r$ is null homotopic then we get a homotopy action of $\mathrm{G}{L}_n(\mathbb{Z}/p^r)$ for some $r$. After one suspension this allows one to split $X^n$ using idempotents of ${𝔽}_p{\mathrm{GL}}_n(\mathbb{Z}/p)$ which can be lifted to ${𝔽}_p{\mathrm{GL}}_n(\mathbb{Z}/p^r)$. In fact all of this is possible if $X$ is an $H$-space...

Let $G$ be an abelian group, $R$ a commutative ring of prime characteristic $p$ with identity and $R_{t}G$ a commutative twisted group ring of $G$ over $R$. Suppose $p$ is a fixed prime, $G_p$ and $S\left(R_{t}G\right)$ are the $p$-components of $G$ and of the unit group $U\left(R_{t}G\right)$ of $R_{t}G$, respectively. Let $R^{*}$ be the multiplicative group of $R$ and let $f_{\alpha }\left(S\right)$ be the $\alpha$-th Ulm-Kaplansky invariant of $S\left(R_{t}G\right)$ where $\alpha$ is any ordinal. In the paper the invariants $f_n\left(S\right)$, $n\in \mathbb{N}\cup \left\{0\right\}$, are calculated, provided $G_p=1$. Further, a commutative ring $R$ with identity of prime characteristic $p$ is said...