category_of_all_partitions
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category_of_all_partitions [2019/03/03 09:10] d.gromada |
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| ==== Partitions ==== | ==== Partitions ==== | ||
| - | Let $k,l\in\mathbb{N}_0$, by a **partition** of $k$ upper and $l$ lower points we mean a partition of the set $\{1,\dots,k\}\sqcup\{1,\dots,l\}\approx\{1,\dots,k+l\}$, that is, a decomposition of the set of $k+l$ points into non-empty disjoint subsets, called **blocks**. The first $k$ points are called **upper** and the last $l$ points are called **lower**. The set of all partitions on $k$ upper and $l$ lower points is denoted $\Pscr(k,l)$. We denote the union $\Pscr:=\bigcup_{k,l\in\N_0}\Pscr(k,l)$. | + | Let $k,l\in\mathbb{N}_0$, by a [[partition|partition]] of $k$ upper and $l$ lower points we mean herre a partition of the set $\{1,\dots,k\}\sqcup\{1,\dots,l\}\approx\{1,\dots,k+l\}$, that is, a decomposition of the set of $k+l$ points into non-empty disjoint subsets, called **blocks**. The first $k$ points are called **upper** and the last $l$ points are called **lower**. The set of all partitions on $k$ upper and $l$ lower points is denoted $\Pscr(k,l)$. We denote the union $\Pscr:=\bigcup_{k,l\in\N_0}\Pscr(k,l)$. |
| We illustrate partitions graphically by putting $k$ points in one row and $l$ points on another row below and connecting by lines those points that are grouped in one block. All lines are drawn between those two rows. | We illustrate partitions graphically by putting $k$ points in one row and $l$ points on another row below and connecting by lines those points that are grouped in one block. All lines are drawn between those two rows. | ||
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| - $T_{qp}=T_qT_p$ whenever one of the sides makes sense, | - $T_{qp}=T_qT_p$ whenever one of the sides makes sense, | ||
| - $T_{p^*}=T_p^*$. | - $T_{p^*}=T_p^*$. | ||
| + | Note that the proposition is true only if the category parameter $\delta$ indeed coincide with the dimension $N$ of the $\C^N$ vector spaces. | ||
| - | Consider the symmetric group $S_N$ and denote by $u$ its representation by permutation matrices. | + | Consider the symmetric group $S_N$ and denote by $u$ the representation of $S_N$ by permutation matrices. For any $k,l\in\N_0$ denote the intertwiner spaces |
| + | $$\FundRep_{S_N}(k,l):=\Rep_{S_N}(u^{\otimes k},u^{\otimes l}):=\{T\colon(\C^N)^{\otimes k}\to(\C^N)^{\otimes l}\mid Tu^{\otimes k}=u^{\otimes l}T\}.$$ | ||
| - | ==== Tannaka–Krein duality ==== | + | **Proposition.** It holds that |
| + | $$\FundRep_{S_N}(k,l)=\{T_p\mid p\in\Part(k,l)\}.$$ | ||
| + | That is, the functor $T$ maps the category of partitions $\Part$ to the category generated by the fundamental representation $u$. | ||
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| + | So the category of all partitions can be used to model the representation theory of the symmetric group. Note however that the functor $T$ is //not// injective, so it does not provide a category isomorphism. | ||
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| + | Note that this result can be also understood as a generalization of the [[Schur–Weyl duality|Schur–Weyl duality]]. | ||
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| + | ==== Completing the category ==== | ||
category_of_all_partitions.1551604243.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
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