Peter Fiebig
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Peter Fiebig has a total of 2 co-authors. Here are the top co-authors by number of shared publications:
- Martina Lanini
Univ Roma Tor Vergata, Dipartimento Matemat, Via Ric Sci 1, I-00133 Rome, Italy
6 shared publications - Martina Lanini1 shared publication
Latest Publications
- Open Access
Representations and binomial coefficients
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 arithmetic properties of binomial coefficients (such as q-versions of Lucas' theorem and the Pfaff-Saalschütz identity) to construct a Frobenius pull back functor and prove a Steinberg's tensor product formula. Then we show that these results imply the corresponding results for reductive algebraic groups in positive characteristics and for quantum groups at roots of unity.
Open AccessRepresentations and binomial coefficients
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 arithmetic properties of binomial coefficients (such as q-versions of Lucas' theorem and the Pfaff-Saalschütz identity) to construct a Frobenius pull back functor and prove a Steinberg's tensor product formula. Then we show that these results imply the corresponding results for reductive algebraic groups in positive characteristics and for quantum groups at roots of unity.
- Open Access
Quantum tilting modules over local rings
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 ${\mathcal X}={\mathcal X}_{\mathscr A}(R)$ associated with a Noetherian ${\mathbb Z}[v,v^{-1}]$-domain ${\mathscr A}$ and a root system $R$. We show that if ${\mathscr A}$ is of quantum characteristic $0$, the model category contains all $U_{\mathscr A}$-modules that admit a Weyl filtration. If ${\mathscr A}$ is in addition local, we study torsion phenomena in the model category. This leads to a construction of torsion free objects in ${\mathcal X}$. We show that these correspond to tilting modules for the quantum group associated with ${\mathscr A}$ and $R$.Open AccessQuantum tilting modules over local rings
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 ${\mathcal X}={\mathcal X}_{\mathscr A}(R)$ associated with a Noetherian ${\mathbb Z}[v,v^{-1}]$-domain ${\mathscr A}$ and a root system $R$. We show that if ${\mathscr A}$ is of quantum characteristic $0$, the model category contains all $U_{\mathscr A}$-modules that admit a Weyl filtration. If ${\mathscr A}$ is in addition local, we study torsion phenomena in the model category. This leads to a construction of torsion free objects in ${\mathcal X}$. We show that these correspond to tilting modules for the quantum group associated with ${\mathscr A}$ and $R$. - Open Access
- 2 September 2022
- Published by Springer Nature in Transformation Groups
- p. 1-20
PERIODICITY FOR SUBQUOTIENTS OF THE MODULAR CATEGORY 𝒪
- 2 September 2022
- Published by Springer Nature in Transformation Groups
- p. 1-20
Open AccessPERIODICITY FOR SUBQUOTIENTS OF THE MODULAR CATEGORY 𝒪
- Open AccessPeriodicity for subquotients of the modular category $\mathcal{O}$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 $\mathcal{O}_{[\mathcal{K}]}$ of $\mathcal{O}$. It has the property that its simple objects are parametrized by elements in $\mathcal{K}$. We then show that $\mathcal{O}_{[\mathcal{K}]}$ is equivalent to $\mathcal{O}_{[\mathcal{K}+p^l\gamma]}$ for any dominant weight $\gamma$ if $l>0$ is an integer such that $\mathcal{K}\cap (\mathcal{K}+p^l\eta)=\emptyset$ for all dominant weights $\eta$. This allows one, for example, to restrict attention to subquotients inside the dominant (or the antidominant) chamber.Open AccessPeriodicity for subquotients of the modular category $\mathcal{O}$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 $\mathcal{O}_{[\mathcal{K}]}$ of $\mathcal{O}$. It has the property that its simple objects are parametrized by elements in $\mathcal{K}$. We then show that $\mathcal{O}_{[\mathcal{K}]}$ is equivalent to $\mathcal{O}_{[\mathcal{K}+p^l\gamma]}$ for any dominant weight $\gamma$ if $l>0$ is an integer such that $\mathcal{K}\cap (\mathcal{K}+p^l\eta)=\emptyset$ for all dominant weights $\eta$. This allows one, for example, to restrict attention to subquotients inside the dominant (or the antidominant) chamber.
- Open Access
Tilting modules and torsion phenomena
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 generations of characters} philosophy of Lusztig and Lusztig-Williamson.
Open AccessTilting modules and torsion phenomena
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 generations of characters} philosophy of Lusztig and Lusztig-Williamson.
- 1 September 2020
- Published by Springer Nature in Transformation Groups
- Vol. 25 (3) , 725-753
SHEAVES ON THE ALCOVES AND MODULAR REPRESENTATIONS I
- 1 September 2020
- Published by Springer Nature in Transformation Groups
- Vol. 25 (3) , 725-753
SHEAVES ON THE ALCOVES AND MODULAR REPRESENTATIONS I
- 26 August 2020
- Published by Oxford University Press (OUP) in International Mathematics Research Notices
- Vol. 2022 (4) , 3031-3056
Lefschetz Operators, Hodge–Riemann Forms, and Representations
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 graded semisimple representation of the corresponding Lie algebra, if and only if there exists a bilinear form that satisfies properties (roughly) analogous to those of the Hodge–Riemann forms in complex geometry. In the 2nd part of the article, we replace the field by the $p$-adic integers (with $p\ne 2$) and show that in this case the existence of a certain bilinear form is equivalent to the existence of a structure of a tilting module for the associated simply connected $p$-adic Chevalley group.- 26 August 2020
- Published by Oxford University Press (OUP) in International Mathematics Research Notices
- Vol. 2022 (4) , 3031-3056
Lefschetz Operators, Hodge–Riemann Forms, and Representations
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 graded semisimple representation of the corresponding Lie algebra, if and only if there exists a bilinear form that satisfies properties (roughly) analogous to those of the Hodge–Riemann forms in complex geometry. In the 2nd part of the article, we replace the field by the $p$-adic integers (with $p\ne 2$) and show that in this case the existence of a certain bilinear form is equivalent to the existence of a structure of a tilting module for the associated simply connected $p$-adic Chevalley group.- 29 April 2020
- Published by Springer Nature in Transformation Groups
- Vol. 25 (3) , 755-791
SHEAVES ON THE ALCOVES AND MODULAR REPRESENTATIONS II
- 29 April 2020
- Published by Springer Nature in Transformation Groups
- Vol. 25 (3) , 755-791
SHEAVES ON THE ALCOVES AND MODULAR REPRESENTATIONS II
- Open Access
Sheaves on the alcoves and modular representations I
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 wall reflection we associate a wall crossing functor on S. In the companion article "Sheaves on the alcoves and modular representations II" we prove that S encodes the simple rational characters of the connected, simply connected algebraic group with root system R over k, in the case that k is algebraically closed with characteristic above the Coxeter number.
Open AccessSheaves on the alcoves and modular representations I
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 wall reflection we associate a wall crossing functor on S. In the companion article "Sheaves on the alcoves and modular representations II" we prove that S encodes the simple rational characters of the connected, simply connected algebraic group with root system R over k, in the case that k is algebraically closed with characteristic above the Coxeter number.
- Open Access
Sheaves on the alcoves and modular representations II
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 projective objects encode the simple rational characters of a reductive algebraic group in all characteristics above the Coxeter number.
Open AccessSheaves on the alcoves and modular representations II
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 projective objects encode the simple rational characters of a reductive algebraic group in all characteristics above the Coxeter number.