Semisimplicity and Indecomposable Objects in Interpolating Partition Categories

Abstract

We study Karoubian tensor categories, which interpolate representation categories of families of the so-called easy quantum groups in the same sense in which Deligne’s interpolation categories $\ensuremath {\mathop {\textrm {\underline {Rep}}}}(S_t)$ interpolate the representation categories of the symmetric groups. As such categories can be described using a graphical calculus of partitions, we call them interpolating partition categories. They include $\ensuremath {\mathop {\textrm {\underline {Rep}}}}(S_t)$ as a special case and can generally be viewed as subcategories of the latter. Focusing on semisimplicity and descriptions of the indecomposable objects, we prove uniform generalisations of results known for special cases, including $\ensuremath {\mathop {\textrm {\underline {Rep}}}}(S_t)$ or Temperley–Lieb categories. In particular, we identify those values of the interpolation parameter, which correspond to semisimple and non-semisimple categories, respectively, for all the so-called group-theoretical easy quantum groups. A crucial ingredient is an abstract analysis of certain subobject lattices developed by Knop, which we adapt to categories of partitions. We go on to prove a parametrisation of the indecomposable objects in all interpolating partition categories for non-zero interpolation parameters via a system of finite groups, which we associate to any partition category, and which we also use to describe the associated graded rings of the Grothendieck rings of these interpolation categories.

Traces on diagram algebras I: Free partition quantum groups, random lattice paths and random walks on trees

“…Also we remark that by [17, Theorem 5.13], the set of generic parameters (that is parameter values for which $$A_{(\mathcal {C},\delta )}(k)$$ is semisimple for all $$k \geqslant 0$$) is always co‐countable. For more precise results on genericity of the loop parameter, see [11].…”

Traces on diagram algebras II: centralizer algebras of easy groups and new variations of the Young graph

“…The exact set of generic parameter G C values has been determined for all of the above algebras except for the rook-Brauer algebra. Namely, G O = C\Z [42], G S = C\N [24] and G H = C\N [11].…”

“…For the choice δ = n, none of these algebras are semisimple when 2k n, see e.g. [11]. However, whenever one chooses δ "generically" (see Remark 3.6), the diagram algebras above are semisimple for all k 0 and therefore admit infinite versions, that is to say direct limit algebras A (X ,δ) (∞), X = S, O, B, H.…”