We obtain a presentation of the principal subspace of the basic A (2) 2n –module, n ≥ 1. We show that its full character is given by the Nahm sum of the tadpole Dynkin diagram Tn = A2n/Z2. This character is conjecturally modular after certain specializations. We prove the modularity property in the case of the affine Lie algebra of type A (2) 4 .

We show that in a triangulated category, the existence of a cluster tilting object often implies that the homomorphism groups are bounded in size. This holds for the stable module category of a selfinjective algebra, and as a corollary we recover a theorem of Erdmann and Holm. We then apply our result to Calabi-Yau triangulated categories, in particular stable categories of maximal Cohen-Macaul...

0.1. Let G be a connected reductive algebraic group defined over a finite field Fq and let G(Fq) be the finite group of all Fq-rational points of G. We would like to present here a strategy for computing the character table of G(Fq) , under the assumption that p, the characteristic of Fq , is sufficiently large. We can assume that G has a simply connected derived group. Indeed, in the general c...

For a linear semisimple Lie group we obtain a necessary and sufficient condition for a highest weight module with non-singular infinitesimal character to be unitarizable.

For a function φ in L(0, 1), extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates φ(nx), n = 1, 2, 3, . . . , constitutes a Riesz basis or a complete sequence in L(0, 1). The problem translates into a question concerning multipliers and cyclic vectors in the Hilbert space H of Dirichlet series f(s) = ∑ n ann, where the coefficients an ar...

Problems. Problem 1. Let Y be a finite type, separated k-scheme. Let E be a locally free OY -module of rank r + 1. Let πE : PY (E)→ Y, φ : π∗E∨ → O(1) be a universal pair of a morphism to Y together with an invertible quotient of the pullback of E∨ (to help calibrate conventions, this is covariant in E with respect to locally split monomorphisms of locally free sheaves). Recall in the proof of ...

We show that certain canonical realizations of the complexes Hom(G,H) and Hom+(G,H) of (partial) graph homomorphisms studied by Babson and Kozlov are in fact instances of the polyhedral Cayley trick. For G a complete graph, we then characterize when a canonical projection of these complexes is itself again a complex, and exhibit several well-known objects that arise as cells or subcomplexes of ...

(ii) (Lazard’s Theorem) L is the polynomial algebra Z[x1, x2, ...]. We proved Lazard’s theorem last semester. We also proved that MU∗ is isomorphic to Z[x1, x2, ...], so MU∗ and L are both polynomial algebras with generators in even degrees (positive even degrees in MU∗, and negative even degrees in L). The usual complex orientation on MU gives rise to a formal group law, and thus we have a hom...