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Register a new userOn equivariant maps related to the space of pairs of exceptional Jordan algebras
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Let $mathcal{J}$ be the exceptional Jordan algebra and $V=mathcal{J}oplus mathcal{J}$. We construct an equivariant map from $V$ to $mathrm{Hom}_k(mathcal{J}otimes mathcal{J},mathcal{J})$ defined by homogeneous polynomials of degree $8$ such that if $xin V$ is a generic point, then the image of $x$ is the structure constant of the isotope of $mathcal{J}$ corresponding to $x$. We also give an alternative way to define the isotope corresponding to a generic point of $mathcal{J}$ by an equivariant map from $mathcal{J}$ to the space of trilinear forms.
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Given a finite group $G$ and two unitary $G$-representations $V$ and $W$, possible restrictions on Brouwer degrees of equivariant maps between representation spheres $S(V)$ and $S(W)$ are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in $S(V)$ (denoted $alpha(V)$). Effective applications of these congruences is limited by answers to the following questions: (i) under which conditions, is ${alpha}(V)>1$? and (ii) does there exist an equivariant map with the degree easy to calculate? In the present paper, we address both questions. We show that ${alpha}(V)>1$ for each irreducible non-trivial $C[G]$-module if and only if $G$ is solvable. For non-solvable groups, we use 2-transitive actions to construct complex representations with non-trivial ${alpha}$-characteristic. Regarding the second question, we suggest a class of Norton algebras without 2-nilpotents giving rise to equivariant quadratic maps, which admit an explicit formula for the Brouwer degree.
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We determine all values of the parameters for which the cell modules form a standard system, for a class of cellular diagram algebras including partition, Brauer, walled Brauer, Temperley-Lieb and Jones algebras. For this, we develop and apply a general theory of algebras with Borelic pairs. The theory is also applied to give new uniform proofs of the cellular and quasi-hereditary properties of the diagram algebras and to construct quasi-hereditary 1-covers, in the sense of Rouquier, with exact Borel subalgebras, in the sense of Konig. Another application of the theory leads to a proof that Auslander-Dlab-Ringel algebras admit exact Borel subalgebras.
We study quantized Coulomb branches of quiver gauge theories of Jordan type. We prove that the quantized Coulomb branch is isomorphic to the spherical graded Cherednik algebra in the unframed case, and is isomorphic to the spherical cyclotomic rational Cherednik algebra in the framed case. We also prove that the quantized Coulomb branch is a deformation of a subquotient of the Yangian of the affine $mathfrak{gl}(1)$.
In this paper the authors investigate the $q$-Schur algebras of type B that were constructed earlier using coideal subalgebras for the quantum group of type A. The authors present a coordinate algebra type construction that allows us to realize these $q$-Schur algebras as the duals of the $d$th graded components of certain graded coalgebras. Under suitable conditions an isomorphism theorem is proved that demonstrates that the representation theory reduces to the $q$-Schur algebra of type A. This enables the authors to address the questions of cellularity, quasi-hereditariness and representation type of these algebras. Later it is shown that these algebras realize the $1$-faithful quasi hereditary covers of the Hecke algebras of type B. As a further consequence, the authors demonstrate that these algebras are Morita equivalent to Rouquiers finite-dimensional algebras that arise from the category ${mathcal O}$ for rational Cherednik algebras for the Weyl group of type B. In particular, we have introduced a Schur-type functor that identifies the type B Knizhnik-Zamolodchikov functor.
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