Let \(G\) be a group. A linear representation of \(G\) is a pair \((\rho, V)\), where \(V\) is a finite dimensional vector space over \(\mathbb{C}\) and \(\rho: G \to GL(V)\) is a group homomorphism. Without ambiguity, we will call \(\rho\) a representation of \(G\). Let \((\rho_1, V_1)\) and \((\rho_2, V_2)\) be two representations of \(G\), an intertwining operator between them is a linear map \(T: V_1 \to V_2\) such that for any \(g \in G\), the following diagram commutes: \[\begin{equation}\label{eq:diagram} \require{AMScd}\begin{CD} V_1 @>T>> V_2 \\ @V{\rho_1(g)}VV @VV{\rho_2(g)}V \\ V_1 @>T>> V_2 \end{CD} \end{equation}\] Let \(\text{Hom}_G(\rho_1, \rho_2)\) be the space of all intertwining operators between \(\rho_1\) and \(\rho_2\).

Let \((\rho, V)\) be a representation of \(G\). Let \(H\) be a subgroup of \(G\), then we can restrict \(\rho\) to \(H\) to get a representation of \(H\). We will use \(\text{Res}^G_H \, \rho\) to denote the restricted representation of \(H\) from \(\rho\). Conversely, if \(\pi\) is a representation of \(H\), then we can construct a representation of \(G\) from \(\pi\), which is known as induced representation of \(G\) from \(\pi\), denoted as \(\text{Ind}_H^G \, \pi\). In this post, I will first talk about the precise description of induced representations and the relations between \(\text{Res}^G_H\) and \(\text{Ind}_H^G\). I will then discuss Mackey’s theorem, which dictates a further relation between restricted and induced representations.

Let \(A \in End(V)\) and \(B \in End(W)\) be two linear maps. We can define naturally the tensor product \(A \otimes B\) of \(A\) and \(B\), from \(V \otimes W\) to \(V \otimes W\), sending \(v \otimes w\) to \(Av \otimes Bw\). In this post, I am going to realize \(A \otimes B\) as a matrix and relate the determinant and trace of \(A \otimes B\) to the ones of \(A\) and \(B\).

Let \(V\) and \(W\) be two vector spaces over a field \(K\) with \(\dim V=n\) and \(\dim W=m\). Let \(e_1, \dots, e_n\) be a basis of \(V\) and let \(f_1, \dots, f_m\) a basis of \(W\). Under the basis, a linear map \(A: V \to V\) can be realized as an \(n \times n\) matrix \((a_{ij})_{1 \le i,j \le n}\), where \(a_{ij} \in K\). Similarly, a linear map \(B: W \to W\) can be realized as a \(m \times m\) matrix \((b_{kl})_{1 \le k, l \le m}\). Now, \(V \otimes W\), as a vector space, has basis \(\{e_i \otimes f_j: 1 \le i \le n, 1 \le j \le m\}\). And, \[\begin{align*} A \otimes B (e_i \otimes f_j) &= A(e_i) \otimes B(f_j) \\ &= \sum_{k}a_{ki}e_k \otimes \sum_{l}b_{lj}f_l \\ &= \sum_{k,l}a_{ki}b_{lj} e_k \otimes f_l. \end{align*}\]

In a topology group \(G\), an open subgroup \(H\) is also closed. The proof of this statement is not hard: \(G=\bigcup_{g_i} g_iH\) is a disjoint union of open left cosets, where \(\{g_i\}\) is a complete representatives set of \(G/H\). Then \(\bigcup_{g_i \ne 1}g_iH\) is open and \(H=G-\bigcup_{g_i \ne 1}g_iH\) is the complement of an open set, and therefore \(H\) is closed. In this post, I will prove a slightly more general theorem:

We will see that an open subgroup \(H\) is locally closed, so it’s closed by the Theorem. Before the proof of the Theorem, let’s talk about locally closed sets first.

Let \(T\) be an algebraic group. \(T\) is called an algebraic torus over \(k\), if \(T(E)\) is isomorphic to a finite direct product of copies of \(G_m(E)\) for some finite finite extension \(E\) of \(k\), where \(G_m\) is the multiplicative group. If \(E\) can be \(k\), then \(T\) is called a split torus over \(k\); otherwise, \(T\) is called a non split torus over \(k\). In this post, I am going to talk something about \(SO\) to give examples of non split and split tori.

Definition

Let \(\mathbb{R}^{+}\) be the set of positive real numbers. Given a function \(f\) on \(\mathbb{R}^{+}\), define the Mellin transform of \(f\), whenever it makes sense, as follow: \[\mathcal{M}(f)(s)=\int_{0}^{\infty} f(t)t^{s} \frac{dt}{t}. \; (1)\]

The very first example of the Mellin transform I have known is the gamma function, \[\Gamma(s) = \int_{0}^{\infty} e^{-t}t^{s} \frac{dt}{t},\] which is the Mellin transform of the function \(e^{-t}\).

Let’s fix a natural number \(k\). How many ways can we decompose \(n\) into \(k\) squares? It’s an interesting problem and it should help me understand automorphic forms better.

Let \(w_k(n)\) be the number of tuples \((n_1, \cdots, n_k)\in \mathbb{Z}^k\) such that \(n_1^2+\cdots+n_k^2=n\). Let \(g_k(q)\) be the generating function of \(w_k(n)\), i.e., \[g_k(q)=\sum_{n \in \mathbb{N}} w_k(n)q^n.\] Then, \[\begin{split} g_k(q) &= \sum_{n \in \mathbb{N}} w_k(n)q^n \\ &= \sum_{n \in \mathbb{N}} \sum_{\substack{n_1, \cdots, n_k \in \mathbb{Z} \\ n_1^2+\cdots+n_k^2=n}}q^n \\ &= \sum_{n_1, \cdots, n_k \in \mathbb{Z}} q^{n_1^2+\cdots+n_k^2} \\ &= \left(\sum_{n \in \mathbb{Z}} q^{n^2}\right)^k \\ &= (g_1(q))^k. \end{split}\]

I have been suffering from insomnia since I came back to Columbus. I thought it was just a matter of jet lag. But after a week or two, I began to realize that it is something else, something unknown yet, that matters. Insufficient sleep turns me into a zombie around 2 or 3 pm in the afternoon.

I had exactly the same problem last December. I tried many ways and finally figured out alcohol worked it out for me. However, alcohol has lost its power on me this time, soon after I finished my rum. Now I go to swimming and work out every other day to make me exhausted at the end of day. Theoretically, this should make me tired and then I could fall asleep easier. Reality always beats theory. It just worsen the consequence of insomnia.

雨后的倦意与泥土里的水分在蒸发着
或白或黄的土狗们还横七竖八地躺着
两只躲着我的小花猫在高处柔柔咪叫
当然 …

2013年有两件重要的事情：毕业、出国——一个阶段的结束伴随另一个阶段的开始。

毕业

回忆的时候总觉得时间过得很快——不知不觉中，我已经毕业半年了。在那个特殊的季节里，我不断回忆我大学四年所做过的事情。杭州的闷热让我略感烦躁，而烦躁在侵蚀我的记忆。记忆中的大一与日记中的大一出现了很大的偏差，对于很多当时发生的事情，当下的感受比当时的感受淡多了，时间抹去了很多强烈的情绪。大二大三的时候，学习生活的单调性让我不怎么写日记了。没有了日记的对照，大二大三只剩下骨头了。但我相信，大二大三是我大学里最用功的两年。

大四上学期忙完申请，大四下学期就没有多少事情了，花时间的只有一门关于李代数表示的讨论班以及毕业论文。我的毕业论文比较简单：只是说明一些对称恒等式（特别是涉及Schur函数的对称恒等式）以及同构表示之间的对应关系。所以，我在毕业那段时间是很闲的。人太闲会出问题的，闲下来之后，很多让我不能入睡的问题像一颗颗子弹正中红心。毕业那段时间也是我失眠最厉害的一段时间。

1、前一段时间总是凌晨两三点才能入睡。每当我置身于黑暗中，我就开始想这样的一个问题：上帝是否存在？这个问题的起源是几个“为什么”。为什么物体会往下掉？因为有重力。为什么会有重力？因为有质量的物体能产生重力场，从而产生重力。那为什么有质量的物体能产生重力场？以我现在的知识，我无法解答这个为什么。我觉得无论是怎样的“为什么”，不断追问下去的话，最终总会问到”谁创造宇宙“的问题。

朋友给了我这样的一个反问来说明上帝是存在的：

如果你在沙滩上看到一行字，你绝对不会认为这是自然产生的，你肯定是认为是由智慧生物，第一反应当然是人，写上去的。那人类本身呢？

这是一个很好的反问，我是承认的，但不足以说服我。这是一个具有强烈导向性的反问，并没有说明为什么人类不能偶然地产生，或者没有直接的证据说明上帝的存在。