arXiv subjects

Robert Muth

Publications and source records attributed to Robert Muth.

24 records · Page 2Linked to original sources

arXiv2018

Schurifying quasi-hereditary algebras

We define and study new classes of quasi-hereditary and cellular algebras which generalize Turner's double algebras. Turner's algebras provide a local description of blocks of symmetric groups up to derived equivalence. Our general construction allows one to `schurify' any quasi-hereditary algebra $A$ to obtain a generalized Schur algebra $S^A(n,d)$ which we prove is again quasi-hereditary if $d\leq n$. We describe decomposition numbers of $S^A(n,d)$ in terms of those of $A$ and the classical Schur algebra $S(n,d)$. In fact, it is essential to work with quasi-hereditary superalgebras $A$, in which case the construction of the schurification involves a non-trivial full rank sub-lattice $T^A_\mathfrak{a}(n,d)\subseteq S^A(n,d)$.

math.RT↗

arXiv2017

Super RSK correspondence with symmetry

Super RSK correspondence is a bijective correspondence between superbiwords and pairs of semistandard supertableaux. Such a bijection was given by Bonetti, Senato and Venezia, via an insertion algorithm closely related to Schensted insertion. Notably, the symmetry property satisfied by the classical RSK bijection holds only in special cases under this bijection. We present a new super RSK bijection, based on the mixed insertion process defined by Haiman, where the symmetry property holds in complete generality.

math.CO↗

arXiv2017

Affine zigzag algebras and imaginary strata for KLR algebras

KLR algebras of affine ADE types are known to be properly stratified if the characteristic of the ground field is greater than some explicit bound. Understanding the strata of this stratification reduces to semicuspidal cases, which split into real and imaginary subcases. Real semicuspidal strata are well-understood. We show that the smallest imaginary stratum is Morita equivalent to Huerfano-Khovanov's zigzag algebra tensored with a polynomial algebra in one variable. We introduce affine zigzag algebras and prove that these are Morita equivalent to arbitrary imaginary strata if the characteristic of the ground field is greater than the bound mentioned above.

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arXiv2016

Stratifying KLR algebras of affine ADE types

We generalize imaginary Howe duality for KLR algebras of affine ADE types, developed in our previous paper, from balanced to arbitrary convex preorders. Under the assumption that the characteristic of the ground field is greater than some explicit bound, we prove that these KLR algebras are properly stratified.

math.RT↗

arXiv2015

Graded skew Specht modules and cuspidal modules for Khovanov-Lauda-Rouquier algebras of affine type A

Kleshchev, Mathas and Ram (2012) gave a presentation for graded Specht modules over Khovanov-Lauda-Rouquier algebras of finite and affine type A. We show that this construction can be applied more generally to skew shapes to give a presentation of graded skew Specht modules, which arise as subquotients of restrictions of Specht modules. As an application, we show that cuspidal modules associated to a balanced convex preorder in affine type A are skew Specht modules for certain hook shapes.

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arXiv2013

Imaginary Schur-Weyl duality

We study imaginary representations of the Khovanov-Lauda-Rouquier algebras of affine Lie type. Irreducible modules for such algebras arise as simple heads of standard modules. In order to define standard modules one needs to have a cuspidal system for a fixed convex preorder. A cuspidal system consists of irreducible cuspidal modules---one for each real positive root for the corresponding affine root system ${\tt X}_l^{(1)}$, as well as irreducible imaginary modules---one for each $l$-multipartition. We study imaginary modules by means of `imaginary Schur-Weyl duality'. We introduce an imaginary analogue of tensor space and the imaginary Schur algebra. We construct a projective generator for the imaginary Schur algebra, which yields a Morita equivalence between the imaginary and the classical Schur algebra. We construct imaginary analogues of Gelfand-Graev representations, Ringel duality and the Jacobi-Trudy formula.

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