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We prove that every finite dimensional representation of a finite group over a field of characteristic p admits a finite resolution by p-permutation modules. The proof involves a reformulation in terms of derived categories.

We prove that, up to adding a complement, every modular representation of a finite group admits a finite resolution by permutation modules.

We prove that the only separable commutative ring-objects in the stable module category of a finite cyclic p-group G are the ones corresponding to subgroups of G. We also describe the tensor-closure of the Kelly radical of the module…

We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$, the derived and the stable categories…

We establish a derived geometric Satake equivalence for the real group $G_{\mathbb R}=PSO(2n-1,1)$ (resp. $PE_6(F_4)$), to be called the Lorentzian Satake equivalence (resp. Octonionic Satake equivalence). By applying the real-symmetric…

In this paper, Whittaker modules are studied for a subalgebra $\mathfrak{q}_{\epsilon}$ of the $\emph{N}$=2 superconformal algebra. The Whittaker modules are classified by central characters. Additionally, criteria for the irreducibility of…

These are notes of a graduate course on representations of non-compact semisimple Lie groups given by the author at MIT.

Using Nadler's Geometric Satake Equivalence for real reductive groups, we obtain a description of the equivariant homology of the loop space of splitting-rank symmetric spaces in terms of the relative dual group of the space. The…

We investigate the (separated) monomorphism category $\operatorname{mono}(Q,\Lambda)$ of a quiver $Q$ over an Artin algebra $\Lambda$. We construct an epivalence from $\overline{\operatorname{mono}}(Q,\Lambda)$ to…

Let $g$ be a complex semisimple Lie algebra with Weyl group $W$. Let $H(W)$ be the Iwahori-Hecke algebra associated to $W$. For each $w\in W$, let $T_w$ and $C_w$ be the corresponding $Z$-graded twisting functor and $Z$-graded shuffling…

We apply convex geometry (cones, fans) to homological input (abelian categories, hearts of bounded t-structures) to construct a new invariant of an abelian category, its heart fan. This can be viewed as a `universal phase diagram' for…

We find new presentations of the modified Ariki-Koike algebra (known also as Shoji's algebra) $\mathcal H_{n,r}$ over an integral domain $R$ associated with a set of parameters $q,u_1,\ldots,u_r$ in $R$. It turns out that the algebra…

We give a geometric model for any algebraic heart in the derived category of a gentle algebra, which is equivalent to the module category of some gentle algebra. To do this, we deform the geometric model for the module category of a gentle…

We prove that there are no strictly shod algebras in hereditary gentle algebras by geometric models. As an application, we give a classification of the silted algebras for Dynkin type $\mathbb{A}_{n}$ and $\widetilde{\mathbb{A}}_{n}$.

Let $\Q$ be the quiver of Dynkin type $\mathbb{A}_n$ with linear orientation and $A_{n}=k\Q$. In this paper, we give a complete classification of the silted algebras of type $A_{n}$ by using the geometric models of gentle algebras. We show…

We study sheaves of modules for the Lie algebra of vector fields with the action of the algebra of functions, compatible via the Leibniz rule. A crucial role in this theory is played by the virtual jets of vector fields - jets that evaluate…

We conjecture a precise relationship between Lusztig $q$-weight multiplicities for type $C$ and Kirillov-Reshetikhin crystals. We also define $\mathfrak{gl}_n$-version of $q$-weight multiplicity for type $C$ and conjecture the positivity.

In this paper, we consider the mixed tensor space of a $G$-graded vector space where $G$ is a finite abelian group. We obtain a spanning set of invariants of the associated symmetric algebra under the action of a color analogue of the…

A conjecture due to Y. Han asks whether that Hochschild homology groups of a finite dimensional algebra vanish for sufficiently large degrees would imply that the algebra is of finite global dimension. We investigate this conjecture from…

By inventing the notion of honeycombs, A. Knutson and T. Tao proved the saturation conjecture for Littlewood-Richardson coefficients. The Newell-Littlewood numbers are a generalization of the Littlewood-Richardson coefficients. By…