### 数学代写|表示论代写Representation theory代考|MATH5735

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|表示论代写Representation theory代考|Symmetric Group Actions

There is a natural left action of $S_{n}$ on an arbitrary $n$-tuple of objects $\left(\lambda_{1}, \ldots, \lambda_{n}\right)$ :
$$\sigma\left(\lambda_{1}, \ldots, \lambda_{n}\right)=\left(\lambda_{\sigma^{-1}(1)}, \ldots, \lambda_{\sigma^{-1}(n)}\right), \quad \sigma \in S_{n}$$
Also, given $V=V_{0} \oplus V_{\overline{1}}$ a vector superspace with parity $p$, we have a linear left action of the symmetric group $S_{n}$ on the tensor product $V^{\otimes n}\left(\sigma \in S_{n}, v_{1}, \ldots, v_{n} \in\right.$ $V)$ :
$$\sigma\left(v_{1} \otimes \cdots \otimes v_{n}\right):=\epsilon_{v}(\sigma) v_{\sigma^{-1}(1)} \otimes \cdots \otimes v_{\sigma^{-1}(n)}$$
where, following the Koszul-Quillen rule,
$$\epsilon_{v}(\sigma):=\prod_{i\sigma(j)}(-1)^{p\left(v_{i}\right) p\left(v_{j}\right)}$$

In particular, if $V$ is purely even $\epsilon_{v}(\sigma)=1$, while if $V$ is purely odd $\epsilon_{v}(\sigma)=$ $\operatorname{sign}(\sigma)$. The corresponding right action of $S_{n}$ on the space $\operatorname{Hom}\left(V^{\otimes n}, V\right)$ is given by $\left(f \in \operatorname{Hom}\left(V^{8 n}, V\right), \sigma \in S_{n}\right)$ :
$$f^{\sigma}\left(v_{1} \otimes \cdots \otimes v_{n}\right)=f\left(\sigma\left(v_{1} \otimes \cdots \otimes v_{n}\right)\right)$$

## 数学代写|表示论代写Representation theory代考|Composition of Permutations and Shuffles

Let $n \geq 1$ and $m_{1}, \ldots, m_{n} \geq 0$. We introduce the following notation:
$$M_{0}=0 \text { and } M_{i}=\sum_{j=1}^{i} m_{j}, \quad i=1, \ldots, n$$
Given $\sigma \in S_{n}$ and $\tau_{1} \in S_{m_{1}}, \ldots, \tau_{n} \in S_{m_{n}}$, we describe the composition
$$\sigma\left(\tau_{1}, \ldots, \tau_{n}\right) \in S_{M_{n}}$$
by saying how it acts on the tensor power $V^{\otimes M_{n}}$ of a vector space $V$ :
$$\left(\sigma\left(\tau_{1}, \ldots, \tau_{n}\right)\right)\left(v_{1} \otimes \cdots \otimes v_{M_{n}}\right)=\sigma\left(\tau_{1}\left(v_{1} \otimes \cdots \otimes v_{M_{1}}\right) \otimes \cdots \otimes \tau_{n}\left(v_{M_{n-1}+1} \otimes \cdots \otimes v_{M_{n}}\right)\right)$$
Definition 3.1 A permutation $\sigma \in S_{m+n}$ is called an ( $\left.m, n\right)$-shuffle if
$$\sigma(1)<\cdots<\sigma(m), \quad \sigma(m+1)<\cdots<\sigma(m+n)$$
The subset of $(m, n)$-shuffles is denoted by $S_{m, n} \subset S_{m+n}$.
Observe that, by definition, $S_{0, n}=S_{n, 0}={1}$ for every $n \geq 0$. If either $m$ or $n$ is negative, we set $S_{m, n}=\emptyset$ by convention.

## 数学代写|表示论代写Representation theory代考|n-Graphs

For an oriented graph $\Gamma$, we denoted by $V(\Gamma)$ the set of vertices of $\Gamma$, and by $E(\Gamma)$ the set of edges. We call $\Gamma$ an $n$-graph if $V(\Gamma)={1, \ldots, n}$. Denote by $\mathcal{G}(n)$ the set of all $n$-graphs without tadpoles, and by $\mathcal{G}{0}(n)$ the set of all acyclic $n$-graphs. An $n$-graph $L$ will be called an $n$-line, or simply a line, if its set of edges is of the form $\left{i{1} \rightarrow i_{2}, i_{2} \rightarrow i_{3}, \ldots, i_{n-1} \rightarrow i_{n}\right}$, where $\left{i_{1}, \ldots, i_{n}\right}$ is a permutation of ${1, \ldots, n}$.

We have a natural left action of $S_{n}$ on the set $\mathcal{G}(n)$ : for the $n$-graph $\Gamma$ and the permutation $\sigma$, the new $n$-graph $\sigma(\Gamma)$ is defined to be the same graph as $\Gamma$ but with the vertex which was labeled as $i$ relabeled as $\sigma(i)$, for every $i=1, \ldots, n$. So, if the $n$-graph $\Gamma$ has an oriented edge $i \rightarrow j$, then the $n$-graph $\sigma(\Gamma)$ has the oriented edge $\sigma(i) \rightarrow \sigma(j)$. Note that $S_{n}$ permutes the set of $n$-lines.

Let us recall the cocomposition of $n$-graphs, as described in [BDSHK19]. Given an $n$-tuple $\left(m_{1}, \ldots, m_{n}\right)$ of positive integers, let $M_{i}$ be as in (3.5). If $\Gamma \in \mathcal{G}\left(M_{n}\right)$, define $\Delta_{i}^{m_{1}, \ldots, m_{n}}(\Gamma) \in \mathcal{G}\left(m_{i}\right), i=1, \ldots, n$, to be the subgraph of $\Gamma$ associated with the set of vertices $\left{M_{i-1}+1, \ldots, M_{i}\right}$, relabeled as $\left{1, \ldots, m_{i}\right}$. Define also $\Delta_{0}^{m_{1}, \ldots, m_{n}}(\Gamma)$ to be the graph obtained from $\Gamma$ by collapsing the vertices and the edges of each $\Delta_{i}^{m_{1}, \ldots, m_{n}}(\Gamma)$ into a single vertex, relabeled as $i$. Then the cocomposition map is the map
\begin{aligned} \Delta^{m_{1}, \ldots, m_{n}}: \mathcal{G}\left(M_{n}\right) & \rightarrow \mathcal{G}(n) \times \mathcal{G}\left(m_{1}\right) \times \cdots \times \mathcal{G}\left(m_{n}\right) \ \Gamma & \mapsto\left(\Delta_{0}^{m_{1}, \ldots, m_{n}}(\Gamma), \Delta_{1}^{m_{1}, \ldots, m_{n}}(\Gamma), \ldots, \Delta_{n}^{m_{1}, \ldots, m_{n}}(\Gamma)\right) \end{aligned}

## 数学代写|表示论代写Representation theory代考|Symmetric Group Actions

σ(λ1,…,λn)=(λσ−1(1),…,λσ−1(n)),σ∈小号n

σ(在1⊗⋯⊗在n):=ε在(σ)在σ−1(1)⊗⋯⊗在σ−1(n)

ε在(σ):=∏一世σ(j)(−1)p(在一世)p(在j)

Fσ(在1⊗⋯⊗在n)=F(σ(在1⊗⋯⊗在n))

## 数学代写|表示论代写Representation theory代考|Composition of Permutations and Shuffles

σ(τ1,…,τn)∈小号米n

(σ(τ1,…,τn))(在1⊗⋯⊗在米n)=σ(τ1(在1⊗⋯⊗在米1)⊗⋯⊗τn(在米n−1+1⊗⋯⊗在米n))

σ(1)<⋯<σ(米),σ(米+1)<⋯<σ(米+n)

## 数学代写|表示论代写Representation theory代考|n-Graphs

Δ米1,…,米n:G(米n)→G(n)×G(米1)×⋯×G(米n) Γ↦(Δ0米1,…,米n(Γ),Δ1米1,…,米n(Γ),…,Δn米1,…,米n(Γ))

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## MATLAB代写

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