Representations of Brauer category and categorification

We study representations of the locally unital and locally finite dimensional algebra $B$ associated to the Brauer category $\mathcal B(\delta_0)$ with defining parameter $\delta_0$ over an algebraically closed field $K$ with characteristic $p\neq 2$. The Grothendieck group $K_0(B\text{-mod}^\Delta)$ will be used to categorify the integrable highest weight $\mathfrak {sl}_{K}$-module $V(\varpi_{\frac{\delta_0-1}{2}})$ with the fundamental weight $\varpi_{\frac{\delta_0-1}{2}}$ as its highest weight, where $B$-mod$^\Delta$ is a subcategory of $B$-lfdmod in which each object has a finite $\Delta$-flag, and $\mathfrak {sl}_{K}$ is either $\mathfrak{sl}_\infty$ or $\hat{\mathfrak{sl}}_p$ depending on whether $p=0$ or $2\nmid p$. As $\mathfrak g$-modules, $\mathbb C\otimes_{\mathbb Z} K_0(B\text{-mod}^\Delta)$ is isomorphic to $V(\varpi_{\frac{\delta_0-1}{2}})$, where $\mathfrak g$ is a Lie subalgebra of $\mathfrak {sl}_{K}$ (see Definition~4.2). When $p=0$, standard $B$-modules and projective covers of simple $B$-modules correspond to monomial basis and so-called quasi-canonical basis of $V(\varpi_{\frac{\delta_0-1}{2}})$, respectively.Comment: 23 page

Isomorphisms between simple modules of degenerate cyclotomic Hecke algebras

We give explicit isomorphisms between simple modules of degenerate cyclotomic Hecke algebras defined via various cellular bases. A special case gives a generalized Mullineux involution in the degenerate case.Comment: 21 page

Representation type of cyclotomic quiver Hecke algebras of type Aℓ(1)A_\ell^{(1)}

We first investigate a connected quiver consisting of all dominant maximal weights for an integrable highest weight module in affine type A. This quiver provides an efficient method to obtain all dominant maximal weights. Then, we completely determine the representation type of cyclotomic Khovanov-Lauda-Rouquier algebras of arbitrary level in affine type A, by using the quiver we construct. This result gives a complete classification for the representation type of blocks of cyclotomic Hecke algebras since cyclotomic KLR algebras of type $A^{(1)}_\ell$ form a one-parameter family and cyclotomic Hecke algebras occur at a special parameter, i.e., $t=-2$ if $\ell=1$ and $t=(-1)^{\ell+1}$ if $\ell\geq2$.Comment: 62 pages, minor revision, accepted by Advances in Mathematic

The Jucys-Murphy basis and semisimplicty criteria for the qq-Brauer algebra

We construct the Jucys-Murphy elements and the Jucys-Murphy basis for the $q$-Brauer algebra in the sense of Mathas[11]. We also give a necessary and sufficient condition for the $q$-Brauer algebra being (split) semisimple over an arbitrary field.Comment: 21 page