Stability patterns in representation theory

Abstract

We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is an array of equivalences between the stable representation category and various other categories, each of which has its own flavor (representation theoretic, combinatorial, commutative algebraic, or categorical) and offers a distinct perspective on the stable category. We use this theory to produce a host of specific results, e.g., the construction of injective resolutions of simple objects, duality between the orthogonal and symplectic theories, a canonical derived auto-equivalence of the general linear theory, etc.

79 pages; v2: increased font size, added 1.4.7, 3.3.8, Remark 6.2.6, and references

References in corpus (10)

• FI-modules and stability for representations of symmetric groups
• GL-equivariant modules over polynomial rings in infinitely many variables
• Introduction to twisted commutative algebras
• Noetherianity of some degree two twisted commutative algebras
• Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra q(n)
• GL-equivariant modules over polynomial rings in infinitely many variables. II
• Homology of Littlewood complexes
• A New Approach to the Representation Thoery of the Symmetric Groups. 2
• Infinite rank spinor and oscillator representations
• Categories of integrable $sl(\infty )$-, $o(\infty )$-, $sp(\infty )$-modules

Cited by in corpus (49)

• Gröbner methods for representations of combinatorial categories
• GL-equivariant modules over polynomial rings in infinitely many variables
• Representation stability and finite linear groups
• Homology of FI-modules
• Introduction to twisted commutative algebras
• Noetherianity of some degree two twisted commutative algebras
• Noetherianity of some degree two twisted skew-commutative algebras
• Deligne categories and the limit of categories $Rep(GL(m|n))$
• On the cohomology of Torelli groups
• Higher order representation stability and ordered configuration spaces of manifolds
• Topological noetherianity for cubic polynomials
• Representation stability for cohomology of configuration spaces in ${\mathbf{R}}^d$
• Cohomology of generalized configuration spaces
• The representation theory of Brauer categories I: triangular categories
• Hilbert series for twisted commutative algebras
• $\mathrm{VI}$ modules in non-describing characteristic, Part I
• Stability in the high-dimensional cohomology of congruence subgroups
• Border Ranks of Monomials
• Koszulity of directed categories in representation stability theory
• Uniformly Presented Vector Spaces
• Integrable $sl(\infty )$-modules and Category $\mathcal{O}$ for $\mathfrak{gl}(m|n)$
• Infinite rank spinor and oscillator representations
• Sp-equivariant modules over polynomial rings in infinitely many variables
• Jacobi-Trudi determinants and characters of minimal affinizations
• Syzygies of secant ideals of Plücker-embedded Grassmannians are generated in bounded degree
• On the geometry and representation theory of isomeric matrices
• Stable characters from permutation patterns
• Schur-Weyl duality for Deligne categories II: the limit case
• Topological actions of Temperley-Lieb algebras and Representation Stability
• Are all Secant Varieties of Segre Products Arithmetically Cohen-Macaulay?
• Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings
• Homology of analogues of Heisenberg Lie algebras
• Effective and Infinite-Rank Superrigidity in the Context of Representation Stability
• Notes on Restricted Inverse Limits of Categories
• On the growth of the Kronecker coefficients
• A new approach to the representation theory of the partition category
• Some generalizations of Schur functors
• Stable representation theory: beyond the classical groups
• Representation stability for sequences of 0-Hecke modules
• A Survey of Representation Stability Theory
• The semi-linear representation theory of the infinite symmetric group
• Brauer diagrams, modular operads, and a graphical nerve theorem for circuit algebras
• An application of Nakayama functor in representation stability theory
• A remark about 6j symbols and young semi-normal form
• On ideals in U$(\mathfrak{sl}(\infty ))$, U$(\mathfrak{o}(\infty ))$, U$(\mathfrak{sp}(\infty ))$
• Centralizers of the infinite symmetric group
• Sheaves of modules on atomic sites and discrete representations of topological groups
• Asymptotic behaviors of representations of graded categories with inductive functors
• Kronecker Comultiplication of Stable Characters and Restriction From $S_{mn}$ to $S_m\times S_n$