We show that if X is a cocompact G-CW-complex such that each isotropy
subgroup Gσ is L(2)-good over an arbitrary commutative ring k, then X satisfies some fixed-point
formula which is an L(2)-analogue of Brown’s formula in 1982. Using this result we present a fixed
point formula for a cocompact proper G-CW-complex which relates the equivariant L(2)-Euler
characteristic of a fixed point CW-complex Xs and the Euler characteristic of X/G. As corollaries,
we prove Atiyah’s theorem in 1976, Akita’s formula in 1999 and a result of Chatterji-Mislin in
2009. We also show that if X is a free G-CW-complex such that C∗(X) is chain homotopy
equivalent to a chain complex of finitely generated projective Zπ1(X)-modules of finite length and
X satisfies some fixed-point formula over Q or C which is an L(2)-analogue of Brown’s formula, then
χ(X/G)=χ(2)(X). As an application, we prove that the weak Bass conjecture holds for any finitely
presented group G satisfying the following condition: for any finitely dominated CW-complex Y
with π1(Y )=G, Y satisfies some fixed-point formula over Q or C which is an L(2)-analogue of
We prove that the cover ideals of all unimodular hypergraphs have the nonincreasing
depth function property. Furthermore, we show that the index of depth stability of
these ideals is bounded by the number of variables.
We develop a local cohomology theory for FI^m-modules, and show that it in many ways mimics the classical theory for multi-graded modules over a polynomial ring. In particular, we define an invariant of FI^m-modules using this local cohomology theory which closely resembles an invariant of multi-graded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.
We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.
Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FI_G and VI_q, and answer positively an open question of Nagpal on representation stability theory.
In this paper we study representation theory of the category FI^m introduced by Gadish which is a product of copies of the category FI, and show that quite a few interesting representational and homological properties of FI can be generalized to FI^m in a natural way. In particular, we prove the representation stability property of finitely generated FI^m -modules over fields of characteristic 0.
In this paper we describe an inductive machinery to investigate asymptotic behaviors of homology groups and related invariants of representations of certain graded combinatorial categories over a commutative Noetherian ring k, via introducing inductive functors which generalize important properties of shift functors of FI-modules. In particular, a sufficient criterion for finiteness of Castelnuovo-Mumford regularity of finitely generated representations of these categories is obtained. As applications, we show that a few important infinite combinatorial categories appearing in representation stability theory are equipped with inductive functors, and hence the finiteness of Castelnuovo-Mumford regularity of their finitely generated representations is guaranteed. We also prove that truncated representations of these categories have linear minimal resolutions by relative projective modules, which are precisely linear minimal projective resolutions when k is a field of characteristic 0.
We consider the quantum cluster algebras which are injective-reachable and introduce a triangular basis in every seed. We prove that, under some initial conditions, there exists a unique common triangular basis with respect to all seeds. This basis is parametrized by tropical points as expected in the Fock-Goncharov conjecture.
As an application, we prove the existence of the common triangular bases for the quantum cluster algebras arising from representations of quantum affine algebras and partially for those arising from quantum unipotent subgroups. This result implies monoidal categorification conjectures of Hernandez-Leclerc and Fomin-Zelevinsky in the corresponding cases: all cluster monomials correspond to simple modules.
We describe an inductive machinery to prove various properties of representations of a category equipped with a generic shift functor. Specifically, we show that if a property (P) of representations of the category behaves well under the generic shift functor, then all finitely generated representations of the category have the property (P). In this way, we obtain simple criteria for properties such as Noetherianity, finiteness of Castelnuovo-Mumford regularity, and polynomial growth of dimension to hold. This gives a systemetic and uniform proof of such properties for representations of the categories $\FI_G$ and $\OI_G$ which appear in representation stability theory.