Algebras and Representation Theory最新文献

A toric vector bundle (mathcal {E}) is a torus equivariant vector bundle on a toric variety. We give a valuation theoretic and tropical point of view on toric vector bundles. We present three (equivalent) classifications of toric vector bundles, which should be regarded as repackagings of the Klyachko data of compatible (mathbb {Z})-filtrations of a toric vector bundle: (1) as piecewise linear maps to space of (mathbb {Z})-valued valuations, (2) as valuations with values in the semifield of piecewise linear functions, and (3) as points in tropical linear ideals over the semifield of piecewise linear functions. Moreover, we interpret the known criteria for ampleness and global generation of (mathcal {E}) as convexity conditions on its piecewise linear map in (1). Finally, using (2) we associate to (mathcal {E}) a collection of polytopes indexed by elements of a certain (representable) matroid encoding the dimensions of weight spaces of global sections of (mathcal {E}). This recovers and extends the Di Rocco-Jabbusch-Smith matriod and parliament of polytopes of (mathcal {E}). This is a follow up paper to Kaveh and Manon. Math. Zeitschrift 302(3), 1367-1392 (2022).

In this paper we study the structure of finite-dimensional representations of the current Lie algebra of type (A_1), (mathfrak {sl}_2[t]), which are obtained by taking tensor products of local Weyl modules with Demazure modules. We show that such a representation admits a Demazure flag and obtain a closed formula for the graded multiplicities of the level 2 Demazure modules in the filtration of the tensor product of two local Weyl modules for (mathfrak {sl}_2[t]). Furthermore, we show that the tensor product of a local Weyl module with an irreducible (mathfrak {sl}_2[t]) module admits a Demazure filtration and derive the graded character of such tensor product modules. In conjunction with the results of Chari et al. (SIGMA Symmetry Integrability Geom. Methods Appl. 10(032), 2014), our findings provide evidence for the conjecture in Blanton (2017) that the tensor product of Demazure modules of levels m and n respectively has a filtration by Demazure modules of level m + n.

Let (W_{r}(mathbb {F}_{q})) be the ring of Witt vectors of length r with residue field (mathbb {F}_{q}) of characteristic p. In this paper, we study the defining characteristic case of the representations of (textrm{GL}_{n}) and (textrm{SL}_{n}) over the principal ideal local rings (W_{r}(mathbb {F}_{q})) and (mathbb {F}_{q}[t]/t^{r}). Let ({textbf{G}}) be either (textrm{GL}_{n}) or (textrm{SL}_{n}) and F a perfect field of characteristic p, we prove that for most p the group algebras (F[{textbf{G}}(W_{r}(mathbb {F}_{q}))]) and (F[{textbf{G}}(mathbb {F}_{q}[t]/t^{r})]) are not stably equivalent of Morita type. Thus, the group algebras (F[{textbf{G}}(W_{r}(mathbb {F}_{q}))]) and (F[{textbf{G}}(mathbb {F}_{q}[t]/t^{r})]) are not isomorphic in the defining characteristic case.

We combinatorially characterize the number (textrm{cc}_2) of conjugacy classes of involutions in any Coxeter group in terms of higher rank odd graphs. This notion naturally generalizes the concept of odd graphs, used previously to count the number of conjugacy classes of reflections. Moreover, we provide formulae for finite and affine types, besides computing (textrm{cc}_2) for all triangle groups and RACGs.

Two different types of Deligne categories have been defined to interpolate the finite dimensional complex representations of the hyperoctahedral group. The first one, initially defined by Knop and then further studied by Likeng and Savage, uses a categorical analogue of the permutation representation as a tensor generator. The second one, due to Flake and Maassen, is tensor generated by a categorical analogue of the reflection representation. We construct a symmetric monoidal functor between the two and show that it is an equivalence of symmetric monoidal categories.

In this paper, we use Galois descent techniques to find suitable representatives of the regular simple representations of the species of type (2, 2) over (k_n:= k[varepsilon ^{1/n}]), where n is a positive integer and (k:=mathbb {C}(!(varepsilon )!)) is the field of Laurent series over the complexes. These regular representations are essential for the definition of canonical algebras. Our work is inspired by the work done for species of type (1, 4) on k in Geiss and Reynoso-Mercado (Bol. Soc. Mat. Mex. 30(3):87, 2024). We presents all the regular simple representations on the n-crown quiver, and from these, we establish a partial classification of regular simple representations of bimodules type (2, 2).

Let (X=G/H) be a homogeneous spherical variety over an algebraically closed field of characteristic (pge 0). We compute the (p')-parts of (pi _0(H)) and (pi _1(X)) from the spherical system of X.

The aim of this paper is to describe all PBW-deformations of the connected graded ({mathbb {K}})-algebra (mathcal {A}) generated by (x_i, 1le ile n,) with the braiding relations:

Firstly, the complexity (mathcal {C}({mathcal {A}})) of the algebra ({mathcal {A}}) is computed. Then all PBW-deformations of (mathcal {A}) when (nge 2) are given explicitly with the help of the general PBW-deformation theory introduced by Cassidy and Shelton. Finally, it is shown that each non-trivial PBW-deformation of (mathcal {A}) is isomorphic to a Iwahori-Hecke algebra (H_q(n+1)) (of type A) with n generators and an appropriate parameter q. Here, trivial PBW-deformations of ({mathcal {A}}) mean that those PBW-deformations that are isomorphic to ({mathcal {A}}.)

Let (mathfrak {g}) be an affine Lie algebra with index set (varvec{I}) = {0, 1, 2, (ldots , varvec{n}}). It is conjectured that for each Dynkin node (varvec{k} in varvec{I} setminus {{textbf {0}}}) the affine Lie algebra (mathfrak {g}) has a positive geometric crystal. In this paper, we construct a positive geometric crystal for the affine Lie algebra (varvec{D}_{textbf {7}}^{{textbf {(1)}}}) corresponding to the Dynkin spin node (varvec{k}= {textbf {7}}).

Our previous work introduced a category of extended queer crystals, whose connected normal objects have unique highest weight elements and characters that are Schur Q-polynomials. The initial models for such crystals were based on semistandard shifted tableaux. Here, we introduce a simpler construction using certain “primed” decomposition tableaux, which slightly generalize the decomposition tableaux used in work of Grantcharov et al. This leads to a new, shorter proof of the highest weight properties of the normal subcategory of extended queer crystals. Along the way, we analyze a primed extension of Grantcharov et al.’s insertion scheme for decomposition tableaux.