October 7. All talks will be in 325 Behrakis.

    2:00 - 2:30 Hugh Thomas, Generalized Catalan Phenomena via Quiver Representations (joint                       with Colin Ingalls)
   
    2:45 - 3:15 Ralf Schiffler, Cluster categories and duplicated algebras

    3:30 - 4:00 Thomas Bruestle, Are there cluster algebras which are not cluster-tilted?

    4:15 - 4:45 Gordana Todorov, Cluster categories and Nilpotent groups

    5:00 - 5:30 Mark Kleiner, Multiplication maps in the preprojective algebra, the maximal rank                       property, and almost split morpisms (joint with Steven P. Diaz)


October 8. All talks will be in 320 Behrakis.

     9:00 - 10:00 Workshop on matrix problems (additional lecture by Raymundo Bautista)

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    10:15 - 10:45 Natasha Rojkovskaya, Braided central elements

    11:00 - 11:30 Neal W. Stoltzfus, The Positive Root Posets of Coxeter groups and Special Bases                           for the Associated Temperley Lieb algebras (Preliminary Report)

    11:45 - 12:15 Izuru Mori, Symmetry in the Vanishing of Ext and a Conjecture of Auslander

    12:30 -  1:00 Mara Neusel, On the Noether Map

Thomas Bruestle, Are there cluster algebras which are not cluster-tilted?

• We define the concept of a tame cluster algebra and study if there are similar phenomena as for tame finite-dimensional algebras. For instance, it is known that not every tame algebra is tilted, and likewise,
  it turns out that not every cluster algebra comes from a cluster-tilted algebra.

Mark Kleiner, Multiplication maps in the preprojective algebra, the maximal rank property, and almost split morpisms (joint with Steven P. Diaz)

• We show that the multiplication maps in the preprojective algebra satisfy a maximal rank property that is similar to the maximal rank property proven by Hochster and Laksov for the multiplication maps in
  the commutative polynomial ring. The result follows from a more general theorem about maximal rank properties of almost split morphisms and uses a grading of the preprojective algebra previously
  introduced by the second-named author.

Izuru Mori, Symmetry in the Vanishing of Ext and a Conjecture of Auslander

• In this talk, we will find a relationship between two conditions on the vanishing of Ext-groups over an algebra, one is due to Auslander, and the other is due to Avramov and Buchweitz, using Tate-Vogel
  cohomology.

Mara Neusel, On the Noether Map

• Let $\rho: G \lra GL(n,F)$ be a faithful representation of a finite group over a field $F$. Let $V=F^n$. Let
  $V(G)=ind_1^G (V)$ be the to $G$ (co)induced module. We obtain a $G$-equivariant map
  \[
  \F[V(G)] \lra \F[V]
  \]
  between the respective symmetric algebras on the duals. Thus we obtain the Noether map
  \[
  \eta_G^G: \F[V(G)]^G \lra \F[V]^G
  \]
  between the respective rings of invariants. We discuss when this map is surjective, why this
  is interesting, and which conclusions we can draw from that.

Natasha Rojkovskaia, Braided central elements

• We present and study two families of polynomials with coefficients in the center of the universal enveloping algebra, labeled by irreducible rerpesentations of a simple Lie algebra. The polynomials can be viewed as analogues of a determinant and a characteristic polynomial of a  certain non-commutative matrix. We discuss the realtions of the polynomials  to Capelli identities and represernation theory of Yangians.

Ralf Schiffler, Cluster categories and duplicated algebras

• Let A be a hereditary algebra and C_A the cluster category of A. We construct a fundamental domain of C_A inside the left part of the module category of the duplicated  algebra of A. We then give a new realization of the tilting objects in C_A, as tilting modules over the duplicated algebra of A.

Neal W. Stoltzfus, The Positive Root Posets of Coxeter groups and Special Bases for the Associated Temperley Lieb algebras (Preliminary Report)

• Abstract: In previous work (with Josh Genauer) special bases were constructed for the Temperley Lieb algebra utilizing a partial order on a set of restricted sequences. The restricted sequence poset has now been related to the positive root poset of the symmetric group. The original construction can now be systematically generalized to all finite reflection groups.

Hugh Thomas, Generalized Catalan Phenomena via Quiver Representations (joint with Colin Ingalls)

• Associated to any root system is a "generalized Catalan number" which counts several sets of objects associated to the root system. Clusters are one example of such a set; another is the poset of  non-crossing partitions. I will describe the non-crossing partitions associated to an arbitrary root system, and explain a reformulation of them as certain subcategories of the category of representations of a Dynkin quiver. One payoff is that this provides a new proof that the non-crossing partitions in general type form a lattice, which was first proved in a type-free way earlier this year by Brady and Watt. I will also use the representation-theoretic perspective on non-crossing partitions to explain the bijection constructed by Reading between non-crossing partitions and clusters.

Gordana Todorov, Cluster categories and Nilpotent groups

• To each simply laced Dynkin diagram there is associated Cluster category and also a Nilpotent torsion free monomial group. Tilting theory on the cluster category is used to define a simplicial complex which is homeomorphic to (n-1) sphere. The image of the (n-2) skeleton corresponds to the domains of generalized semiinvariants. However, more surprising was that the same picture also appears as an Igusa-Orr picture describing the boundary of the set of all simples in their complex which is used to compute homology of the group. We are trying to prove that the other pictures for the monomial groups have a similar description.