Robinson-Schensted-Knuth correspondence in the geometry of partial flag varieties

In this paper we generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence.Comment: 10 pages, replaced to include an extra reference that also proves lemma 2.

Towers of graded superalgebras categorify the twisted Heisenberg double

We show that the Grothendieck groups of the categories of finitely-generated graded supermodules and finitely-generated projective graded supermodules over a tower of graded superalgebras satisfying certain natural conditions give rise to twisted Hopf algebras that are twisted dual. Then, using induction and restriction functors coming from such towers, we obtain a categorification of the twisted Heisenberg double and its Fock space representation. We show that towers of wreath product algebras (in particular, the tower of Sergeev superalgebras) and the tower of nilcoxeter graded superalgebras satisfy our axioms. In the latter case, one obtains a categorification of the quantum Weyl algebra.Comment: 29 pages; v2: Minor changes (corrected typos, etc.) throughou

Fixed rings of twisted generalized Weyl algebras

Twisted generalized Weyl algebras (TGWAs) are a large family of algebras that includes several algebras of interest for ring theory and representation theory, such as Weyl algebras, primitive quotients of $U(\mathfrak{sl}_2)$, and multiparameter quantized Weyl algebras. In this work, we study invariants of TGWAs under diagonal automorphisms. Under certain conditions, we are able to show that the fixed ring of a TGWA by such an automorphism is again a TGWA. In particular, this is true for $\Bbbk$-finitistic TGWAs of type $(A_1)^n$ and $A_2$. We apply this theorem to study properties of the fixed ring, such as the noetherian property and simplicity. We also look at the behavior of simple weight modules for TGWAs when restricted to the action of the fixed ring. As an auxiliary result, in order to study invariants of tensor products of TGWAs, we prove that the class of regular, $\mu$-consistent TGWAs is closed under tensor products

Twists of twisted generalized Weyl algebras

We study graded twisted tensor products and graded twists of twisted generalized Weyl algebras (TGWAs). We show that the class of TGWAs is closed under these operations assuming mild hypotheses. We generalize a result on cocycle equivalence amongst multiparameter quantized Weyl algebras to the setting of TGWAs. As another application we prove that certain TGWAs of type $A_2$ are noetherian.Comment: 15 page