Algebras and Representation Theory最新文献

We use a unified elementary approach to prove the second part of classical, mixed, super, and mixed super Schur-Weyl dualities for general linear groups and supergroups over an infinite ground field of arbitrary characteristic. These dualities describe the endomorphism algebras of the tensor space and mixed tensor space, respectively, over the group algebra of the symmetric group and the Brauer wall algebra, respectively. Our main new results are the second part of the mixed Schur-Weyl dualities and mixed super Schur-Weyl dualities over an infinite ground field of positive characteristic.

Let (mathscr {A}) be a connected cochain DG algebra such that (H(mathscr {A})) is a Noetherian graded algebra. We give some criteria for (mathscr {A}) to be homologically smooth in terms of the singularity category, the cone length of the canonical module k and the global dimension of (mathscr {A}). For any cohomologically finite DG (mathscr {A})-module M, we show that it is compact when (mathscr {A}) is homologically smooth. If (mathscr {A}) is in addition Gorenstein, we get

where (textrm{CMreg}M) is the Castelnuovo-Mumford regularity of M, (textrm{depth}_{mathscr {A}}mathscr {A}) is the depth of (mathscr {A}) and ( mathrm {Ext.reg}, M) is the Ext-regularity of M.

We find new presentations of the modified Ariki-Koike algebra (known also as Shoji’s algebra) (mathcal {H}_{n,r}) over an integral domain R associated with a set of parameters (q,u_1,ldots ,u_r) in R. It turns out that the algebra (mathcal {H}_{n,r}) has a set of generators (t_1,ldots ,t_n) and (g_1,ldots g_{n-1}) subject to some defining relations similar to the relations of Yokonuma-Hecke algebra. We also obtain a presentation of (mathcal {H}_{n,r}) which is independent of the choice of (u_1,ldots u_r). As applications of the presentations, we find an explicit and direct isomorphism between the modified Ariki-Koike algebras with different choices of parameters ((u_1,ldots ,u_r)). We also find an explicit trace form on the algebra (mathcal {H}_{n,r}) which is symmetrizing provided the parameters (u_1,ldots , u_r) are invertible in R. We show that the symmetric group (mathfrak {S}(r)) acts on the algebra (mathcal H_{n,r}), and find a basis and a set of generators of the fixed subalgebra (mathcal H_{n,r}^{mathfrak {S}(r)}).

We show that the order type of the simplest version of a hammock for string algebras lies in the class of finite description linear orders–the smallest class of linear orders containing (textbf{0}), (textbf{1}), and that is closed under isomorphisms, finite order sum, anti-lexicographic product with (omega ) and (omega ^*), and shuffle of finite subsets–using condensation (localization) of linear orders as a tool. We also introduce two finite subsets of the set of bands and use them to describe the location of left (mathbb {N})-strings in the completion of hammocks.

To every Hopf heap or quantum cotorsor of Grunspan a Hopf algebra of translations is associated. This translation Hopf algebra acts on the Hopf heap making it a Hopf-Galois co-object. Conversely, any Hopf-Galois co-object has the natural structure of a Hopf heap with the translation Hopf algebra isomorphic to the acting Hopf algebra. It is then shown that this assignment establishes an equivalence between categories of Hopf heaps and Hopf-Galois co-objects.

Let H be a finite-dimensional Hopf algebra over an algebraically closed field (Bbbk ) with the dual Chevalley property. We prove that H is of finite corepresentation type if and only if it is coNakayama, if and only if the link quiver (textrm{Q}(H)) of H is a disjoint union of basic cycles, if and only if the link-indecomposable component (H_{(1)}) containing (Bbbk 1) is a pointed Hopf algebra and the link quiver of (H_{(1)}) is a basic cycle.

We prove that the duals of the quantum Frobenius morphisms and their splittings by Lusztig are compatible with quantum cluster monomials. After specialization, we deduce that the canonical Frobenius splittings on flag varieties are compatible with cluster algebra structures on Schubert cells.

In this paper, we first study the center of a quotient of the reduced enveloping algebra of p(n), then we use the obtained results to investigate the restricted representation of p(n).

Let (chi ) be an irreducible character of a group G,  and (S_c(G)=sum _{chi in textrm{Irr}(G)}textrm{cod}(chi )) be the sum of the codegrees of the irreducible characters of G. Write (textrm{fcod} (G)=frac{S_c(G)}{|G|}.) We aim to explore the structure of finite groups in terms of (textrm{fcod} (G).) On the other hand, we determine the lower bound of (S_c(G)) for nonsolvable groups and prove that if G is nonsolvable, then (S_c(G)geqslant S_c(A_5)=68,) with equality if and only if (Gcong A_5.) Additionally, we show that there is a solvable group so that it has the codegree sum as (A_5.)

We show that a categorical action of shifted 0-affine algebra naturally gives two families of pairs of complementary idempotents in the triangulated monoidal category of triangulated endofunctors for each weight category. Consequently, we obtain two families of pairs of complementary idempotents in the triangulated monoidal category ({textrm{D}}^btextrm{Coh}(G/P times G/P)). As an application, this provides examples where the projection functors of a semiorthogonal decomposition are kernel functors, and we determine the generators of the component categories in the Grassmannians case.