Mathematics

We complete the discrete cluster categories of type A as defined by Igusa and Todorov, by embedding such a discrete cluster category inside a larger one, and then taking a certain Verdier quotient. The resulting category is a Hom-finite Krull-Schmidt triangulated category containing the discrete cluster category as a full subcategory. The objects and Hom-spaces in this new category can be described geometrically, even though the category is not 2 -Calabi-Yau and Ext-spaces are not always symmetric. We describe all cluster-tilting subcategories. Given such a subcategory, we define a cluster character that takes values in a ring with infinitely many indeterminates. Our cluster character is new in that it takes into account infinite dimensional sub-representations of infinite dimensional ones. We show that it satisfies the multiplication formula and also the exchange formula, provided that the objects being exchanged satisfy some local Calabi-Yau conditions.

We produce minimal integrity bases for both isotropic and hemitropic invariant algebras (and more generally covariant algebras) of most common bidimensional constitutive tensors and -- possibly coupled -- laws, including piezoelectricity law, photoelasticity, Eshelby and elasticity tensors, complex viscoelasticity tensor, Hill elasto-plasticity, and (totally symmetric) fabric tensors up to twelfth-order. The concept of covariant, which extends that of invariant is explained and motivated. It appears to be much more useful for applications. All the tools required to obtain these results are explained in detail and a cleaning algorithm is formulated to achieve minimality in the isotropic case. The invariants and covariants are first expressed in complex forms and then in tensorial forms, thanks to explicit translation formulas which are provided. The proposed approach also applies to any n -uplet of bidimensional constitutive tensors.

Two matrix vector spaces V,W⊂ C n×n are said to be equivalent if SVR=W for some nonsingular S and R . These spaces are congruent if R= S T . We prove that if all matrices in V and W are symmetric, or all matrices in V and W are skew-symmetric, then V and W are congruent if and only if they are equivalent. Let F:U×⋯×U→V and G: U ′ ×⋯× U ′ → V ′ be symmetric or skew-symmetric k -linear maps over C . If there exists a set of linear bijections φ 1 ,…, φ k :U→ U ′ and ψ:V→ V ′ that transforms F to G , then there exists such a set with φ 1 =⋯= φ k .

We introduce an analytic method that uses the global dimension function gldim to produce contractible flows on the space StabD of stability conditions on a triangulated category D . In the case when D=D( S λ ) is the topological Fukaya category of a graded surface S λ , we show that gldim −1 (0,y) contracts to gldim −1 (0,x) for any 1≤x≤y , provided (x,y) does not contain `critical' values {1+ w ∂ / m ∂ ∣ w ∂ ≥0,∂∈∂ S λ } , where the pair ( m ∂ , w ∂ ) consists of the number m ∂ of marked points and the winding number w ∂ associated to a boundary component ∂ of S λ . One consequence is that the global dimension of D( S λ ) must be one of these critical values. Besides, we remove the assumptions in Kikuta-Ouchi-Takahashi's classification result on triangulated categories with global dimension less than 1.

Let G be a semisimple simply connected complex algebraic group. Let U be the unipotent radical of a Borel subgroup in G . We describe the coordinate rings of U (resp., G/U , G ) in terms of two (resp., four, eight) birational charts introduced in [L94, L19] in connection with the study of total positivity.

We study corners and fundamental corners of the irreducible representations of the groups G=Spin(n,1) that are not elementary, i.e. that are equivalent to subquotients of reducible nonunitary principal series representations. For even n we obtain results in a way analogous to the results in [10] for the groups SU(n,1). Especially, we again get a bijection between the nonelementary part G ^ 0 of the unitary dual G ^ and the unitary dual K ^ . In the case of odd n we get a bijection between G ^ 0 and a tru subset of K ^ .

Let F be a field which is, either local non archimedean, or finite, of residual charcateristic p but of characteristic different from 2 . Let W be a symplectic space of finite dimension over F . Suppose R is a field of characteristic ℓ≠p so that there exists a non trivial smooth additive character ψ:F→ R × . Then the Stone-von Neumann theorem of the Heisenberg group H(W) is still valid for representations with coefficients in R . It leads to a projective representation of the group Sp(W) which lifts to a genuine smooth representation of a central extension of Sp(W) by R × : this is the modular Weil representation of the metaplectic group. For any dual pair ( H 1 , H 2 ) , their lifts to the metaplectic group may splitor not according to the different cases at stake. Eventually, computing the biggest isotypic quotient of the modular Weil representation allows to define the Θ -lift. Some new lines of investigation are thus available with these new tools such as studying scalar extension and reduction modulo ℓ .

For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are spherical for the action of a Levi subgroup. We evidence the conjecture, employing the combinatorics of Demazure modules, and work of R. Avdeev-A. Petukhov, M. Can-R. Hodges, R. Hodges-V. Lakshmibai, P. Karuppuchamy, P. Magyar-J. Weyman-A. Zelevinsky, N. Perrin, J. Stembridge, and B. Tenner. In type A, we establish connections with the key polynomials of A. Lascoux- M.-P. Schützenberger, multiplicity-freeness, and split-symmetry in algebraic combinatorics. Thereby, we invoke theorems of A. Kohnert, V. Reiner-M. Shimozono, and C. Ross-A. Yong.

Frieze patterns are defined by objects of a category of Dyck paths, to do that, it is introduced the notion of diamond of Dynkin type An. Such diamonds constitute a tool to build integral frieze patterns.

The affine group schemes of automorphisms of the multilinear r-fold cross products on finite-dimensional vectors spaces over fields of characteristic not two are determined. Gradings by abelian groups on these structures, that correspond to morphisms from diagonalizable group schemes into these group schemes of automorphisms, are completely classified, up to isomorphism.