category_of_all_partitions
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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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| The set of all natural numbers with zero $\N_0$ as a set of objects together with the spaces of linear combinations 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 $*$-category. All objects in the category are self-dual. This category is called the **(linear) category of all partitions**. | The set of all natural numbers with zero $\N_0$ as a set of objects together with the spaces of linear combinations 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 $*$-category. All objects in the category are self-dual. This category is called the **(linear) category of all partitions**. | ||
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| + | ===== Representation theory of the symmetric group ===== | ||
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| + | ==== Linear maps associated to partitions ==== | ||
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| + | Suppose the parameter $\delta$ is now a natural number $N\in\N$. For every partition $p\in\Pscr(k,l)$ we define a linear map $T_p\colon(\C^N)^{\otimes k}\to(\C^N)^{\otimes l}$ as | ||
| + | $$T_p(e_{i_1}\otimes\cdots\otimes e_{i_k})=\sum_{j_1,\dots,j_l=1}^N\delta_p(\mathbf{i},\mathbf{j})(e_{j_1}\otimes\cdots\otimes e_{j_l}),$$ | ||
| + | where $\mathbf{i}=(i_1,\dots,i_k)$, $\mathbf{j}=(j_1,\dots,j_l)$ and the symbol $\delta_p(\mathbf{i},\mathbf{j})$ is defined as follows. Let us assign the $k$ points in the upper row of $p$ by the numbers $i_1,\dots,i_k$ (from left to right) and the $l$ points in the lower row by $j_1,\dots,j_l$ (again from left to right). Then $\delta(\mathbf{i},\mathbf{j})=1$ if the points belonging to the same block are assigned the same numbers. Otherwise $\delta(\mathbf{i},\mathbf{j})=0$. | ||
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| + | As an example, we can express $\delta_p$ and $\delta_q$, where $p$ and $q$ come from the example above, using multivariate $\delta$ function as follows | ||
| + | $$\delta_p(\mathbf{i},\mathbf{j})=\delta_{i_1i_2i_3j_2j_3},\quad | ||
| + | \delta_q(\mathbf{i},\mathbf{j})=\delta_{i_2j_3j_4}\delta_{i_3j_2}.$$ | ||
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| + | We extend this definition to the linear spaces $\Part_N(k,l)$ linearly, i.e. $\delta_{\alpha p+q}=\alpha\delta_p+\delta_q$ and hence $T_{\alpha p+q}=\alpha T_p+T_q$. | ||
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| + | ==== Intertwiners for the symmetric group ==== | ||
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| + | **Proposition.** The map $T_\bullet\colon p\mapsto T_p$ is a monoidal unitary functor. That is, we have the following | ||
| + | - $T_{p\otimes q}=T_p\otimes T_q$, | ||
| + | - $T_{qp}=T_qT_p$ whenever one of the sides makes sense, | ||
| + | - $T_{p^*}=T_p^*$. | ||
| + | Note that the proposition is true only if the category parameter $\delta$ indeed coincide with the dimension $N$ of the $\C^N$ vector spaces. | ||
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| + | 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 | ||
| + | $$\FundRep_{S_N}(k,l):=\Rep_{S_N}(u^{\otimes k},u^{\otimes l}):=\{T\colon(\C^N)^{\otimes k}\to(\C^N)^{\otimes l}\mid Tu^{\otimes k}=u^{\otimes l}T\}.$$ | ||
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| + | **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 ==== | ||
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category_of_all_partitions.1551601405.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
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