User Tools

Site Tools

Trace: • compact_matrix_quantum_group • category_of_partitions
category_of_partitions

This is an old revision of the document!

Table of Contents

Category of partitions

This page is about easy categories of partitions in the sense of Banica and Speicher. If we equip those with a linear structure, we get linear categories of partitions.

Definition

The set of all partitions $\Pscr$ can be given the structure of a monoidal involutive category by introducing the following operations.

  • The tensor product of two partitions $p\in\Pscr(k,l)$ and $q\in\Pscr(k',l')$ is the partition $p\otimes q\in \Pscr(k+k',l+l')$ obtained by writing the graphical representations of $p$ and $q$ “side by side”.

$$ \BigPartition{ \Pblock 0 to 0.25:2,3 \Pblock 1 to 0.75:1,2,3 \Psingletons 0 to 0.25:1,4 \Pline (2.5,0.25) (2.5,0.75) } \otimes \BigPartition{ \Psingletons 0 to 0.25:1,4 \Psingletons 1 to 0.75:1,4 \Pline (2,0) (3,1) \Pline (3,0) (2,1) \Pline (2.75,0.25) (4,0.25) } = \BigPartition{ \Pblock 0 to 0.25:2,3 \Pblock 1 to 0.75:1,2,3 \Psingletons 0 to 0.25:1,4,5,8 \Psingletons 1 to 0.75:5,8 \Pline (2.5,0.25) (2.5,0.75) \Pline (6,0) (7,1) \Pline (7,0) (6,1) \Pline (6.75,0.25) (8,0.25) } $$

  • For $p\in\Pscr(k,l)$, $q\in\Pscr(l,m)$ we define their composition $qp\in\Pscr(k,m)$ by putting the graphical representation of $q$ below $p$ identifying the lower row of $p$ with the upper row of $q$. The upper row of $p$ now represents the upper row of the composition and the lower row of $q$ represents the lower row of the composition.

$$ \BigPartition{ \Psingletons 0 to 0.25:1,4 \Psingletons 1 to 0.75:1,4 \Pline (2,0) (3,1) \Pline (3,0) (2,1) \Pline (2.75,0.25) (4,0.25) } \cdot \BigPartition{ \Pblock 0 to 0.25:2,3 \Pblock 1 to 0.75:1,2,3 \Psingletons 0 to 0.25:1,4 \Pline (2.5,0.25) (2.5,0.75) } = \BigPartition{ \Pblock 0.5 to 0.75:2,3 \Pblock 1.5 to 1.25:1,2,3 \Psingletons 0.5 to 0.75:1,4 \Pline (2.5,0.75) (2.5,1.25) \Psingletons -0.5 to -0.25:1,4 \Psingletons 0.5 to 0.25:1,4 \Pline (2,-0.5) (3,0.5) \Pline (3,-0.5) (2,0.5) \Pline (2.75,-0.25) (4,-0.25) } = \BigPartition{ \Pblock 0 to 0.25:2,3,4 \Pblock 1 to 0.75:1,2,3 \Psingletons 0 to 0.25:1 \Pline (2.5,0.25) (2.5,0.75) } $$

  • For $p\in\Pscr(k,l)$ we define its involution $p^*\in\Pscr(l,k)$ by reversing its graphical representation with respect to the horizontal axis.

$$ \left( \BigPartition{ \Pblock 0 to 0.25:2,3 \Pblock 1 to 0.75:1,2,3 \Psingletons 0 to 0.25:1,4 \Pline (2.5,0.25) (2.5,0.75) } \right)^* = \BigPartition{ \Pblock 1 to 0.75:2,3 \Pblock 0 to 0.25:1,2,3 \Psingletons 1 to 0.75:1,4 \Pline (2.5,0.25) (2.5,0.75) } $$

The set of all natural numbers with zero $\N_0$ as a set of objects together with the sets of partitions $\Part_\delta(k,l)$ as sets of morphisms between $k\in\N_0$ and $l\in\N_0$ with respect to those operations form a monoidal involutive category. All objects in the category are self-dual. This category is called the category of all partitions.

Any monoidal involutive subcategory with duals $\Cscr$ is called a category of partitions. That is, a category of partitions is a collection of subsets $\Cscr(k,l)\subset\Pscr(k,l)$ containing the identity partition $\idpart\in\Cscr(1,1)$ and the pair partition $\pairpart\in\Cscr(0,2)$, which is closed under the category operations.

For given $p_1,\dots,p_n\in\Pscr$, we denote by $\langle p_1,\dots,p_n\rangle$ the smallest linear category of partitions containing $p_1,\dots,p_n$. We say that $p_1,\dots,p_n$ generate $\langle p_1,\dots,p_n\rangle$. Note that the pair partition is contained in the category by definition and hence will not be explicitly listed as a generator. Any element in $\langle p_1,\dots,p_n\rangle$ can be obtained from the generators $p_1,\dots,p_n$ and the pair partition $\pairpart$ by performing a finite amount of category operations and linear combinations.

Relation with linear categories

Consider a category of partitions $\Cscr$. Put $\Kscr(k,l):=\spanlin\Cscr(k,l)\subset\Part_\delta(k,l)$. Then $\Kscr$ is a linear category of partitions. Moreover, supposing $\delta\neq 0$, if $\Cscr=\langle p_1,\dots,p_n\rangle$, then $\Kscr=\langle p_1,\dots,p_n\rangle_\delta$.

Conversely, a linear category of partitions $\Kscr$ is called easy if it is spanned by partitions. That is, if there is a collection of sets $\Cscr(k,l)\subset\Pscr(k,l)$ such that $\Kscr(k,l)=\spanlin\Cscr(k,l)$. Then the collection $\Cscr(k,l)$ is a category of partition.

This means that categories of partitions can be understood as an easy subclass of linear categories of partitions. Their advantage is that they are much easier to work with. In particular, a complete classification of categories is available, which serves as a source of many examples of linear categories of partitions. Since any linear category of partitions induces a compact matrix quantum group through the Tannaka–Krein duality, categories of partitions can also serve as a source of many examples of quantum groups. Quantum groups corresponding to categories of partitions are called easy.

Classification of categories of partitions

In this section, we summarize the classification results for categories of partitions. More information is provided in the separate articles.

Non-crossing partitions

Categories containing the crossing partition

Categories containing the half-liberating partition

The hyperoctahedral case

category_of_partitions.1551710230.txt.gz · Last modified: 2021/11/23 11:56 (external edit)

Page Tools

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
CC Attribution-Share Alike 4.0 International Donate Powered by PHP Valid HTML5 Valid CSS Driven by DokuWiki