Problem 1.

Problem 1. Let $G$ be a finite group. Show that $f \in L^{2}(G)$ satisfies $f(g)=1 /|G|$ for all $g \in G$ if and only if the Fourier transform satisfies
$f^{F}(X)=0$ for all non-trivial irreducible matrix representations $X$ of $G$,
and
$f^{F}\left(X_{\text {triv }}\right)=1$ where $X_{\text {triv }}$ is the 1-dimensional trivial matrix representation of $G$.
(Hint: use Fourier inversion to show one direction. To show the other, demonstrate that it is sufficient to prove this for only unitary representations $X$ and then use Schur orthagonality.)

Problem 2. Problem 2. A function $p \in L^{2}(G)$ is said to be a probability measure on $G$ if $p(g) \in[0,1]$ for all $g \in G$ and $\sum_{g \in C} p(g)=1$. The function $u \in L^{2}(G)$ defined by $u(g)=1 /|G|$ for all $g \in G$ is said to be the uniform probability measure on $G$.

Using the definition of group convolution, prove that for any finite group $G$, we have $u * u=u$.
(It is not required in order to solve the problem, but this can be interpreted probabilistically: if $h$ and $k$ are random elements of the group chosen according to the uniform probsbility measure, then this result says that the product $h k$ will be uniformly distributed in the group.)

\begin{prob}

Problem 3. Show that if $p$ is any probability measure on $\mathbb{Z}(N)$ and $p * p=u$, then we must have $p=u$, where as above $u$ is the uniform probability measure on $\mathbb{Z}(N)$.
(Hint: take the Fourier transform of $p+p$ with respect to the 1-dimensional irreducible representations of $\mathbb{Z}(N)$. Show that $p^{F}(X)=0$ for $X$ irreducible unless $X$ is the trivial representation.)
(It is not required in order to solve the problem, but this can be interpreted probabilistically: you will have shown for any probsbility measure $p$ on $\mathbb{Z}(N)$ which is not uniform to begin with, if $a$ and $b$ are randomly chosen aocording to $p$ then the sum $a+b$ will also not be uniform.)

Problem 3.

Problem 4. Let $p$ be the probability measure on $S_{3}$ given by $p(\varepsilon)=1 / 6, p((12))=1 / 6, p((13))=2 / 9, p((23))=1 / 9, p((123))=2 / 9, p((132))=1 / 9 .$
a) Compute $p^{F}\left(X_{1}\right), p^{F}\left(X_{2}\right)$, and $p^{F}\left(X_{3}\right)$, where $X_{1}$ is the 1-dimensional trivial representation of $S_{3}, X_{2}$ is the 1-dimensionsl sign representation of $S_{3}$, and $X_{3}$ is the 2-dimensional standard representation computed in Problem 1b) of Homework 2 .
b) Show that $p * p=u$ where $u$ is the uniform probsbility messure on $S_{\mathrm{s}}$. (Hint: use the computation in part a) and Problem 1.)
c) Say a sentence about why the proof in Problem 3 does not apply for $S_{3}$.
(It is not required in order to solve the problem, but this can be interpreted probabilistically: you’ve found a probability messure $p$ on $S_{3}$ which is not uniform, but such that if $\pi$ and $\sigma$ are chosen according to $p$, the composition $\pi \sigma$ will be uniformly distributed. Said another way: there is a random shuffle which is not itself uniform, but if the random shuffle is done twice the result is uniform!)

Problem 3.

Problem 5. Let $M^{\lambda}$ be the permutation module associated to a partition $\lambda$.
a) Show that the characters of $M^{\lambda}$ are integer-valued functions for all partitions $\lambda$.
b) Show that every irreducible character of $S_{n}$ is an integer-valued function. (Hint: consider the modules $M^{\lambda}$ in turm, with $\lambda$ taken in reverse lexicographic order. Use the decomposition of $M^{\lambda}$ into a direct sum of Specht modules, with $S^{l}$ appearing eractly once in the direct sum.)

Problem 3.

Problem 6. Rocall the notation that a partition $\lambda=\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{\ell}\right)$ can be written $\lambda=$ $1^{m_{1}} 2^{m_{2}} \ldots n^{m_{n}}$ if $\lambda$ has $m_{1}$ parts equal to $1, m_{2}$ parts equal to 2 , etc. For $\lambda \vdash n$ let $\phi^{\lambda}$ be the character of the permutation module $M^{\lambda}$ and let $\pi \in S_{n}$ have cycle type $\lambda$. Show that
$$\phi^{\lambda}(\pi)=m_{1} ! m_{2} ! \cdots m_{n} \text { ! }$$
(Hint: this can be done directly by reasoning combinatorially, but it can also be done using the formula proved in class for the character of an induced representation. You may use in your homework the fact that that the permutation module $M^{\lambda}$ corresponds to the trivial representation of the subgroup $S_{\lambda}=S_{\left[1, \ldots, \lambda_{1}\right]} \times \cdots \times S_{\left[\lambda_{1}+\ldots+\lambda_{k-1}+1, \ldots, \lambda_{1}+\ldots+\lambda_{\varepsilon}\right]}$ inducod to $S_{n} ;$ see Sec. 2.1 of Sagan.)

Problem 3.

Problem 7. a) Compute the partition $\lambda$ and the Young tableaux $(P, Q)$ obtained by applying the Robinson-Schensted algorithm to the permutation $\sigma=654839217 \in S_{9}$ (written in one-line notation).
b) Apply the inverse Robinson-Schensted slgorithm to the pair of Young tablesux
$$P=\begin{array}{|l|l|l|l|} \hline 1 & 4 & 7 & 8 \ \hline 2 & 6 & 9 & \ \hline 3 & & \ \cline { 1 } 5 & & & \ \hline \end{array}$$
to find the associated permutation $\sigma \in S_{g}$. (Write $\sigma$ in one-line notation.)

\end{prob}

BS equation代写

## MATH 251, Lie Groups, Spring 2022:

Office hours:: MW2-3 and by appointment

Office: APM 5256, tel. 534-2734

Email: [email protected]

Prerequisites: A solid understanding and familiarity with basic concepts in algebra (groups, homomorphisms etc) and analysis (convergence, norms) as well as a good understanding of linear algebra. Ask me if in doubt.

Material: This is a continuation of Lie Groups 251B, taught in Winter 2022. We will first concentrate on the representation theory of semisimple Lie agebras. This will include a proof of the character formula, decomposition of tensor products of representations and duality theorems (such as Schur-Weyl duality, Howe duality). Further topics may include real non-compact Lie groups, symmetric spaces and q-deformations of universal enveloping algebras, known as quantum groups. There will not be a fixed course book. The books and lecture notes listed below should cover most of the material of the course. For material not covered in these books, we plan to make other resources available.

Some books/lecture notes related to the course:

Lecture notes for Math 251B Winter 2022

Borcherds’ lecture notes

Ziller’s lecture notes

Lie Groups, Lie Algebras, And Representations : An Elementary Introduction, Brian C. Hall (electronic copy available from our library)

Introduction to Lie Algebras and Representation Theory, James E. Humphreys, Springer (electronic copy available from our library)

Representation Theory. A First Course, Graduate Texts in Mathematics 129, Joe Harris and William Fulton, Springer (electronic copy available from our library)

Some lecture notes for this course:

Lecture 2 March 30

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Please ignore material below the line for now:

Homework 1

Homework 2

Homework 3

Solutions of some homework problems

Proof of Lie product formula

Homework 4

Lecture notes: complete reducibility

Below are some notes for certain topics of the course

Exponential map for matrix Lie groups

Matrix Lie groups are Lie groups

basic properties of tori

Here are some problems and remarks concerning this course:

Problems related to exponential map