Schur Functors and Schur Complexes (1982)
K. Akin, D. A. Buchsbaum, J. Weyman
Abstract: The study of Schur functors has a relatively long history. Its main impetus derived from representation theory, originally in characteristic zero. Over the years, however, with the development of modular representations, and algebraic geometry over fields of positive characteristic, the need for a theory of universal polynomial functors increased and, since the mid-1960s, approaches to a characteristic-free treatment of Schur functors have been developing (see, for instance, the recent book of Green in which the treatments by Carter and Lusztig, Higman, and Towber, among others, are discussed). Our own interest in such a treatment was awakened by the work of Lascoux on resolutions of determinantal ideals. Although his thesis treated only the characteristic zero case, it suggested that a general and elementary theory of Schur functors could be developed using only the rudiments of multilinear algebra (involving the Hopf algebra structures of the symmetric, exterior, and divided power algebras).
The Equations of Conjugacy Classes of Nilpotent Matrices (1989)
Abstract: Let X be the set of n• n matrices over a field k of characteristic 0. For a partition u=(nl, u2..... us) of n we denote by O (u) the set of nilpotent matrices in X with Jordan blocks of sizes ul,..., us. We are interested in equations of the closure of O (u) in Xi. e. in the generators of the ideal of polynomial functions on Xvanishing on O (u). For u=(n), O (u) is the set of all nilpotent matrices and an old result of Kostant proved in the fundamental paper [K] says that the equations are the GL (n)-invariants in the coordinate ring of X (GL (n) acts on X by conjugation). The problem of calculating the equations of O (u) in general was proposed by DeConcini and Procesi in [DP] where the authors calculated the generators of ideals of schematic intersections O (u) c~ D (D is the set of diagonal matrices). DeConcini and Procesi, Tanisaki [T] and Eisenbud and Saltman [ES] proposed different sets of generators of the ideals of O (u).
Semi-Invariants of Quivers and Saturation for Littlewood-Richardson Coefficients (2000)
H. Derksen, J. Weyman
Abstract: Let Q be a quiver without oriented cycles. For a dimension vector Beta let Rep(Q,Beta) be the set of representations of Q with dimension vector Beta. The group GL(Q, Beta) acts on Rep(Q,Beta). In this paper we show that the ring of semi-invariants SI(Q,Beta) is spanned by special semi-invariants c^V associated to representations V of Q. From this we show that the set of weights appearing in SI(Q,Beta) is saturated.
Quivers with Potentials and Their Representations I: Mutations (2008)
H. Derksen, J. Weyman, A. Zelevinsky
Abstract: We study quivers with relations given by noncommutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein–Gelfand–Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi–Yau algebras, cluster algebras.
Quivers with Potentials and Their Representations II: Applications to Cluster Algebras (2010)
H. Derksen, J. Weyman, A. Zelevinsky
Abstract: We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the “Cluster algebras IV” paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a family of integer polynomials called F-polynomials. In the case of skew-symmetric exchange matrices we find an interpretation of these g-vectors and F-polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about g-vectors and F-polynomials made in loc. cit.
Generic Free Resolutions and Root Systems (2018)
Abstract: In this paper I give an explicit construction of the generic rings R-gen for free resolutions of length 3 over Noetherian commutative C-algebras. The key role is played by the defect Lie algebra introduced earlier on. The defect algebra turns out to be a parabolic subalgebra in a Kac–Moody Lie algebra associated to the graph T-p,q,r corresponding to the format of the resolution. The ring R-gen is Noetherian if and only if the graph T-p,q,r corresponding to a given format is a Dynkin diagram. In such case R-gen has rational singularities so it is Cohen–Macaulay. The ring R-gen is a deformation of a commutative ring R-spec which has a structure of a multiplicity free module over a product of Kac–Moody Lie algebras corresponding to the graph T-p,q,r and a product of two general linear Lie algebras.
- J. Wacławek Prize of Institute of Mathematics of Polish Academy of Sciences for an outstanding Ph.D. thesis, 1981
- Prize of Polish Mathematical Society for young mathematicians, for papers published in years 1979-1982, 1982
- Prize of the Polish Mathematical Society for young mathematicians, 1982
- M. Wacławek Prize of the Institute of Polish Academy of Sciences, 1982
- Kuratowski Prize of Polish Mathematical Society (joint with Piotr Pragacz), 1983
- Humboldt Forschungspreis, March 2012
- Simons Research Professor, MSRI, January-May, 2013
- Wacław Sierpiński Medal and Lecture, Polish Mathematical Society and Warsaw University, 2015
- Banach Prize of Polish Mathematical Society for scientific achievment, 2021
- NSF grant DMS-8702809 Research in Commutative Algebra and Invariant Theory, 1989
- NSF grant DMS-8903466 Research in Commutative Algebra, 1991
- NSF grant DMS-9102432 Mathematical Sciences: Research in Commutative Algebra, 1994
- NSF grant DMS-9403703 Syzygies of Special Varieties, 1997
- NSF grant DMS-9700884 Varieties related to Algebraic Group Actions, 2000
- NSF grant DMS-0070658 Applications of Representations of Quivers, 2003
- NSF grant DMS-0300064 Applications of Quiver Representations, 2006
- NSF grant DMS-0600229 Geometric aspects of Quiver Representations, 2009
- NSF grant DMS-0901185, 2014
- NAWA POWROTY – PPN/PPO/2018/1/00013/U/00001 – Applications of Lie algebras to Commutative Algebra, 2018
- OPUS NCN grant UMO-2018/29/BST1/01290, 2018
- NSF grant DMS-1400740, 2018
- MAESTRO NCN – UMO-2019/34/A/ST1/00263 – Research in Commutative Algebra and Representation Theory, 2019
- NSF grant DMS-1802067, 2021