Joseph Chuang

Feynman diagrams and minimal models for operadic algebras.

We construct an explicit minimal model for an algebra over the cobar-construction of a differential graded operad. The structure maps of this minimal model are expressed in terms of sums over decorated trees. We introduce the appropriate notion of a homotopy equivalence of operadic algebras and show that our minimal model is homotopy equivalent to the original algebra. All this generalizes and gives a conceptual explanation of well-known results for A∞-algebras. Furthermore, we show that these results carry over to the case of algebras over modular operads; the sums over trees get replaced by sums over general Feynman graphs. As a by-product of our work we prove gauge-independence of Kontsevichs ‘dual construction’ producing graph cohomology classes from contractible differential graded Frobenius algebras.

Filtrations in Rouquier blocks of symmetric groups and Schur algebras

We study Rouquier blocks of symmetric groups and Schur algebras in detail, and obtain explicit descriptions for the radical layers of the principal indecomposable, Weyl, Young and Specht modules of these blocks. At the same time, the Jantzen filtrations of the Weyl modules are shown to coincide with their radical filtrations. We also address the conjectures of Martin, Lascoux–Leclerc–Thibon–Rouquier and James for these blocks.

SYMMETRIC GROUPS, WREATH PRODUCTS, MORITA EQUIVALENCES, AND BROUÉ'S ABELIAN DEFECT GROUP CONJECTURE

It is shown that for any prime p, and any non-negative integer w less than p, there exist p-blocks of symmetric groups of defect w, which are Morita equivalent to the principal p-block of the group Sp [rmoust ] Sw. Combined with work of J. Rickard, this proves that Broues abelian defect group conjecture holds for p-blocks of symmetric groups of defect at most 5.

Representations of wreath products of algebras

Filtrations of modules over wreath products of algebras are studied and corresponding multiplicity formulas are given in terms of Littlewood–Richardson coefficients. An example relevant to Jantzen filtrations in Schur algebras is presented.

Combinatorics and Formal Geometry of the Maurer–Cartan Equation

We give a general treatment of the Maurer–Cartan equation in homotopy algebras and describe the operads and formal differential geometric objects governing the corresponding algebraic structures. We show that the notion of Maurer–Cartan twisting is encoded in certain automorphisms of these universal objects.

Abstract Hodge Decomposition and Minimal Models for Cyclic Algebras

We show that an algebra over a cyclic operad supplied with an additional linear algebra datum called Hodge decomposition admits a minimal model whose structure maps are given in terms of summation over trees. This minimal model is unique up to homotopy.

Row and column removal in the q-deformed Fock space

Analogues of Jamess row and column removal theorems are proved for the q-decomposition numbers arising from the canonical basis in the q-deformed Fock space.

Runner Removal Morita Equivalences

Let \({U}_{q} = {U}_{q}({\mathfrak{g}\mathfrak{l}}_{n})\) be Lusztig’s divided power quantum general linear group over the complex field with parameter q, a root of unity. We investigate the category of finite-dimensional modules over U q . Lusztig’s famous character formula for simple modules over U q is written purely in terms of the affine Weyl group and its Hecke algebra, which are independent ofq. Our result may be viewed as the categorical version of this independence. Moreover, our methods are valid over fields of positive characteristic. The proof uses the modular representation theory of symmetric groups and finite general linear groups, and the notions of sl2-categorification and perverse equivalences.

Kleshchev's decomposition numbers and branching coefficients in the Fock space

We give combinatorial descriptions of some coefficients of the canonical basis of the q-deformed Fock space representation of Uq(sle) and of some matrix entries for the action of the Chevellay generators fr with respect to the canonical basis. These are q-analogues of results of Kleshchev on decomposition numbers and branching coefficients for symmetric groups and Schur algebras.

Representations of Finite Groups

We provide a formal framework for the theory of representations of finite groups, as modules over the group ring. Along the way, we develop the general theory of groups (relying on the group add class for the basics), modules, and vector spaces, to the extent required for theory of group representations. We then provide formal proofs of several important introductory theorems in the subject, including Maschke’s theorem, Schur’s lemma, and Frobenius reciprocity. We also prove that every irreducible representation is isomorphic to a submodule of the group ring, leading to the fact that for a finite group there are only finitely many isomorphism classes of irreducible representations. In all of this, no restriction is made on the characteristic of the ring or field of scalars until the definition of a group representation, and then the only restriction made is that the characteristic must not divide the order of the group.