In this talk, I will introduce the notion of functorial equivalence of blocks of finite groups, developed in recent joint work with Deniz Yilmaz. For a commutative ring R, and a field k of characteristic p>0, we introduce the category of diagonal p-permutation functors over R. To a pair (G,b) of a finite group G and a block idempotent b of kG, we associate a diagonal p-permutation functor FG,b, and we say that two such pairs (G,b) and (H,c) are functorially equivalent over R if the functors FG,b and FH,c are isomorphic. We show that the category of diagonal p-permutation functors over an algebraically closed field of characteristic 0 is semisimple. We obtain a precise description of the simple functors, and explicit formulas for their multiplicities as summands of FG,b. It follows that functorial equivalence preserves the defect groups of blocks, their number of simple modules, and their number of ordinary irreducible characters. This also leads to characterizations of nilpotent blocks, and to a finiteness theorem in the spirit of Donovan's finiteness conjecture.
The Picard group Tk(G) of the stable module category of a finite group has been an important object of study in modular representation theory, starting with work Dade in the 1970s. Its elements are equivalence classes of so-called endotrivial modules, i.e., modules M such that End(M) is isomorphic to a trivial module direct sum a projective kG-module. 1-dimensional characters, and their shifts, are examples of such modules, but often exotic elements exist as well. My talk will be a guided tour of how to calculate Tk(G), using methods from homotopy theory. The tour will visit joint work with Tobias Barthel and Joshua Hunt, with Jon Carlson, Nadia Mazza and Dan Nakano, and with Achim Krause.
Spin representations of the symmetric group Sn can be thought of equivalently as either projective representations of Sn, or as linear representations of a double cover Sn+ of Sn. Whilst the linear representation theory of Sn is dictated by removing 'rim-hooks' from (the Young diagrams of) partitions of n, the projective representation theory of Sn is controlled by removing ‘bars’ from bar partitions of n (i.e. partitions of n into distinct parts). We will look at some combinatorial results on bar partitions from a recent paper of the author before discussing methods for determining the modular decomposition of spin representations over fields of positive characteristic.
The McKay conjecture asserts that a finite group has the same number of odd degree irreducible characters as the normalizer of a Sylow 2-subgroup. The Alperin-McKay (A-M) conjecture generalizes this to the height-zero characters in 2-blocks.
In his original paper, McKay already showed that his conjecture holds for the finite symmetric groups Sn. In 2016, Giannelli, Tent and the speaker established a canonical bijection realising A-M for Sn; the height-zero irreducible characters in a 2-block are naturally parametrized by tuples of hooks whose lengths are certain powers of 2, and this parametrization is compatible with restriction to an appropriate 2-local subgroup.
Now corresponding to a 2-block of the symmetric group Sn, there is a 2-block of a maximal Young subgroup of Sn of the same weight. An obvious question is whether our canonical bijection is compatible with restriction of height-zero characters between these blocks.
Attempting to prove this compatibility lead me to formulate a conjecture asserting the Schur-positivity of certain differences of skew-Schur functions. The corresponding skew-shapes have triangular inner-shape, but otherwise do not refer to the 2-modular theory. I will describe my conjecture and give positive evidence in its favour.
In a recent paper, joint with Tobias Barthel and Drew Heard, we develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensor-ideals of certain categories of spectral G-Mackey functors for all finite groups G. In this talk, I will provide an introduction to the problem of classifying thick and localizing tensor-ideals via theories of support, describe in broad strokes some of the highlights of our theory (which builds on the work of Balmer-Favi, Stevenson, and Benson-Iyengar-Krause) and, time-permitting, discuss our applications in equivariant homotopy theory. The starting point for these equivariant applications is a recent computation (joint with Irakli Patchkoria and Christian Wimmer) of the Balmer spectrum of the category of derived Mackey functors. We similarly study the Balmer spectrum of the category of E(n)-local spectral Mackey functors, and harness our geometric theory of stratification to classify the localizing tensor-ideals of both categories.
We work in the group algebra kG of a finite group scheme defined over a field of characteristic p>0. The stable category stmod(kG) of finitely generated kG-modules is a tensor triangulated category of compact objects. Associated to any thick tensor ideal subcategory C is a distinguished triangle
→ E → k → F →
where E and F are idempotent modules in stable category StMod(kG) of all kG-modules. Tensoring with F is the localization functor associated to C. Indeed, the endomorphism ring of F is the endomorphism ring of the trivial module in the localized category. In this lecture we will discuss some results on the nature of the endomorphism ring of the module F and some strange variant support varieties for modules.
We work in the group algebra kG of a finite group scheme defined over a field of characteristic p > 0. The stable category stmod(kG) of finitely generated kG-modules is tensor triangulated category of compact objects. Associated to any thick tensor ideal subcategory 𝒞 is a distinguished triangle → E → k → F → where E and F are idempotent modules in the stable category StMod(kG) of all kG-modules. Tensoring with F is the localization functor associated to 𝒞. Indeed, the endomorphism ring of F is the endomorphism ring of the trivial module in the localized category. In this lecture, we will discuss some results on the nature of the endomorphism ring of the module F and some strange variant support varieties for modules.
In a recent paper, Jeremy Rickard showed that for any finite-dimensional algebra R over a field, if the R-injectives generate the derived category D(R), then the finitistic dimension of R is finite (recall that the famous finitistic dimension conjecture claims that for any R that is a finite-dimensional algebra over a field, the finitistic dimension of R should be finite). In the same paper, it was noted that there are no known finite-dimensional algebras over fields for which this injective generation property is absent. In this talk, we will consider the case when R is a group algebra (not necessarily a finite-dimensional algebra over a field), and show that for a large class of groups, the finiteness of the finitistic dimension of the group algebra (over any commutative ring of finite global dimension) implies the above injective generation property. We will also show how this question, for group algebras, is very closely connected to some existing conjectures on various cohomological invariants for groups, and that will lead us to a version of the finitistic dimension conjecture for group algebras.
Suppose that G is a finite group and k is a field of characteristic p>0. Let M be a finitely generated kG-module. We consider the complete cohomology ring ℰ̂M∗=Σn∈ℤ Ex̂tkGn(M, M). We show that the ring has two distinguished ideals I∗ ⊆ J∗ ⊆ ℰ̂M∗ such that I∗ is bounded above in degrees, ℰ̂M∗/J∗ is bounded below in degree and J∗/I∗ is eventually periodic with terms of bounded dimension. We prove that if M is neither projective nor periodic, then the subring of all elements in negative degrees in ℰ̂M∗ is a nilpotent algebra.