Peter Fiebig
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View article: Representations and binomial coefficients
Representations and binomial coefficients Open
To a root system and a choice of coefficients in a field we associate a category of graded spaces with operators . For an arbitrary choice of coefficients we show that we obtain a semisimple category in which the simple objects are para…
View article: Representations and binomial coefficients
Representations and binomial coefficients Open
For a root system R, a field K and a "choice of coefficients in K" we define a category of graded spaces with operators and study some of its properties. Then we assume that the coefficients are given by quantum binomials. We use basic ari…
View article: Quantum tilting modules over local rings
Quantum tilting modules over local rings Open
We show that tilting modules for quantum groups over local Noetherian domains of quantum characteristic 0 exist and the indecomposable tilting modules are parametrized by their highest weight. For this, we introduce a model category associ…
View article: PERIODICITY FOR SUBQUOTIENTS OF THE MODULAR CATEGORY 𝒪
PERIODICITY FOR SUBQUOTIENTS OF THE MODULAR CATEGORY 𝒪 Open
In this paper we study the category 𝒪 over the hyperalgebra of a reductive algebraic group in positive characteristics. For any locally closed subset K of weights, we define a subquotient 𝒪 [ K ] of 𝒪. It has the property that its simple o…
View article: Quantum tilting modules over local rings
Quantum tilting modules over local rings Open
We show that tilting modules for quantum groups over local Noetherian domains exist and that the indecomposable tilting modules are parametrized by their highest weight. For this we introduce a model category ${\mathcal X}={\mathcal X}_{\m…
View article: Periodicity for subquotients of the modular category $\mathcal{O}$
Periodicity for subquotients of the modular category $\mathcal{O}$ Open
In this paper we study the category $\mathcal{O}$ over the hyperalgebra of a reductive algebraic group in positive characteristics. For any locally closed subset $\mathcal{K}$ of weights we define a subquotient $\mathcal{O}_{[\mathcal{K}]}…
View article: Periodicity of irreducible modular and quantum characters
Periodicity of irreducible modular and quantum characters Open
For a root system R, a field K and an invertible element q in K let U be the associated quantum group, defined via Lusztig's divided powers construction. We study the irreducible characters of this algebra with integral (but not necessaril…
View article: Tilting modules and torsion phenomena
Tilting modules and torsion phenomena Open
We construct families of representations for quantum groups over $\mathbb{Z}[v,v^{-1}]$-algebras that interpolate between Weyl modules and tilting modules. These families might be candidates for objects with characters satisfying the {\em …
View article: Approximations of tilting modules
Approximations of tilting modules Open
For any subset $\Theta$ of the natural numbers and any dominant weight $\lambda$ we construct an indecomposable module $S_\Theta(\lambda)$ with maximal weight $\lambda$ for the quantum group $U_{{\mathscr Z}_{\mathfrak p}}$, where ${\maths…
View article: Lefschetz Operators, Hodge–Riemann Forms, and Representations
Lefschetz Operators, Hodge–Riemann Forms, and Representations Open
For a field of characteristic $\ne 2$, we study vector spaces that are graded by the weight lattice of a root system and are endowed with linear operators in each simple root direction. We show that these data extend to a weight lattice gr…
View article: Moment graphs in representation theory and geometry
Moment graphs in representation theory and geometry Open
This paper reviews the moment graph technique that allows to translate certain representation theoretic problems into geometric ones. For simplicity we restrict ourselves to the case of semisimple complex Lie algebras. In particular, we sh…
View article: Sheaves on the alcoves and modular representations II
Sheaves on the alcoves and modular representations II Open
We relate the category of sheaves on alcoves that was constructed in "Sheaves on the alcoves and modular representations I" to the representation theory of reductive algebraic groups. In particular, we show that its indecomposable projecti…
View article: Sheaves on the alcoves and modular representations I
Sheaves on the alcoves and modular representations I Open
We consider the set of affine alcoves associated with a root system R as a topological space and consider a certain category S of sheaves of Z-modules on this space. Here Z is the structure algebra of the root system over a field k. To any…