Narrow Results

In this paper, we define an explicit basis for the ${𝔤𝔩}_n$-web algebra $H_n(k)$ (the ${𝔤𝔩}_n$ generalization of Khovanov’s arc algebra) using categorified $q$-skew Howe duality.

Our construction is a ${𝔤𝔩}_n$-web version of Hu–Mathas’ graded cellular basis and has two major applications: it gives rise to an explicit isomorphism between a certain idempotent truncation of a thick calculus cyclotomic KLR algebra and $H_n(k)$, and it gives an explicit graded cellular basis of the $2$-hom space between two ${𝔤𝔩}_n$-webs. We use this to give a (in principle) computable version of colored Khovanov–Rozansky ${𝔤𝔩}_n$-link homology, obtained from a complex defined purely combinatorially via the (thick cyclotomic) KLR algebra and needs only $F$.

The paper aims to introduce the cyclotomic $q$-Schur superalgebra via the permutation supermodules of the cyclotomic Hecke algebra and investigate its structure. In particular, we show that the cyclotomic $q$-Schur superalgebra is a cellular superalgebra and establish the double centralizer property between the cyclotomic Hecke algebra and the cyclotomic $q$-Schur superalgebra.

In this paper, we give a concrete method to construct cellular algebras from matrix algebras by specifying certain fixed matrices for the data of inflations. In particular, orthogonal matrices can be chosen for such data.