﻿

## Math620 Representation theory课程简介

This course is an introduction to the representation theory of groups. Although the catalog title refers to finite groups, we will consider both finite and infinite groups. Representation theory studies the way in which a given group may act on vector spaces; in other words, it is concerned with representing groups as groups of matrices. We are mostly interested in irreducible representations, which are the building blocks for the construction of all representations of a given group. The main questions in representation theory are related to: (1) the construction of irreducible representations; (2) the calculation of certain algebraic invariants of these; (3) the decomposition of other representations into irreducibles.

Representation theory is a fundamental tool for studying group symmetry – geometric, analytic, or algebraic – by means of linear algebra. Its origins are mostly in the work of F. Frobenius, H. Weyl, I. Schur, and A. Young, about a century ago; Weyl’s work, for instance, is a milestone in the representation theory of Lie groups, which play a central role in many areas of mathematics. Important advances have been made during last century, through the study of representations of more and more general groups, and through a better understanding of the subtle combinatorics involved, which led to some very explicit constructions and computations. Today, representation theory plays an important role in many recent developments of mathematics and theoretical physics. A considerable amount of recent work is devoted to the representation theory of quantum groups, which are certain deformations of Lie groups, with applications to physics.

## PREREQUISITES

Here are some pretty pictures that will come up in this course, created by John Stembridge; they illustrate the hyperplane arrangements corresponding to the root systems A3 and B3 .

Tentative syllabus: The main concepts of representation theory (characters, induced representations, irreducible representations). Symmetric functions (Young tableaux, Schur functions). Representations of the symmetric group (construction of the irreducible representations, Frobenius’s formula). Representations of the general linear group (Weyl’s construction, characters). Extension of Weyl’s construction to other Lie groups and Lie algebras. The main concepts of Lie theory and the classification of complex semisimple Lie algebras. Representations of complex simple Lie groups. The Weyl character formula. A hint about what lies ahead.

Prerequisites for this course are Math 220 (Linear Algebra) and Math 327 (Elementary Abstract Algebra). On the other hand, Math 420 (Abstract Algebra) and Math 424 (Advanced Linear Algebra) are helpful, but not required.

## Math620 Representation theory HELP（EXAM HELP， ONLINE TUTOR）

1. (1) (5 pts) Let $G$ be a finite group. Show that the function
2. 3. \begin{aligned} 4. \mathbb{C}[G] \times \mathbb{C}[G] & \longrightarrow \mathbb{C} \ 5. \left(f_1, f_2\right) & \longmapsto\left\langle f_1, f_2\right\rangle=\frac{1}{|G|} \sum_{g \in G} f_1(g) \overline{f_2(g)} 6. \end{aligned} 7.
8. defines an inner product on $\mathbb{C}[G]$.
9. Solution: We have
10. 11. \begin{aligned} 12. \left\langle c_1 f_1+c_2 f_2, f_3\right\rangle & =\frac{1}{|G|} \sum_{g \in G}\left(c_1 f_1+c_2 f_2\right)(g) \overline{f_3(g)} \ 13. & =\frac{1}{|G|} \sum_{g \in G}\left(c_1 f_1(g) \overline{f_3(g)}+c_2 f_2(g) \overline{f_3(g)}\right) \ 14. & =\frac{1}{|G|} \sum_{g \in G} c_1 f_1(g) \overline{f_3(g)}+\frac{1}{|G|} \sum_{g \in G} c_2 f_2(g) \overline{f_3(g)} \ 15. & =c_1 \frac{1}{|G|} \sum_{g \in G} f_1(g) \overline{f_3(g)}+c_2 \frac{1}{|G|} \sum_{g \in G} f_2(g) \overline{f_3(g)} \ 16. & =c_1\left\langle f_1, f_3\right\rangle+c_2\left\langle f_2, f_3\right\rangle 17. \end{aligned} 18.
19. Also
20. 21. \begin{aligned} 22. \overline{\left\langle f_1, f_2\right\rangle} & =\overline{\frac{1}{|G|} \sum_{g \in G} f_1(g) \overline{f_2(g)}} \ 23. & \frac{1}{|G|} \sum_{g \in G} \overline{f_1(g) \overline{f_2(g)}} \ 24. & \frac{1}{|G|} \sum_{g \in G} \overline{f_1(g)} \overline{\overline{f_2(g)}} \ 25. & \frac{1}{|G|} \sum_{g \in G} \overline{f_1(g)} f_2(g)=\left\langle f_2, f_1\right\rangle 26. \end{aligned} 27.
28. Lastly, we have
29. $$30. \langle f, f\rangle=\frac{1}{|G|} \sum_{g \in G} f(g) \overline{f(g)}=\frac{1}{|G|} \sum_{g \in G}|f(g)|^2 . 31.$$
32. This sum is greater than or equal to zero and is zero if and only if $f(g)=0$ for all $g \in G$.

Therefore, $\langle f, f\rangle=0$ if and only if $f(g)=0$ for all $g \in G$, which implies that $\langle \cdot, \cdot\rangle$ is positive definite. Thus, $\langle \cdot, \cdot\rangle$ is an inner product on $\mathbb{C}[G]$.

1. (2) Instead of taking the trace of $\phi_g$ to define the character, one might try to do the same by taking the determinant of $\phi_g$ instead. This problem shows that this is not as useful since such a character would tell us nothing about non-abelian simple groups (and these are important).
2. For $\phi$ a representation of a finite group $G$, define a function
3. 4. \begin{aligned} 5. \operatorname{det} \phi: G & \longrightarrow \mathbb{C}^{\times} \ 6. g & \longmapsto(\operatorname{det} \phi)(g)=\operatorname{det}\left(\phi_g\right) 7. \end{aligned} 8.
9. (a) (4 pts) Show det $\phi$ is a representatation (and hence it’s a character since characters are same as representations for one-dimensional representations).
10. (b) (5 pts) Show that if $G$ is a non-abelian simple group, then det $\phi$ is the trivial character.
11. Solution:
12. (a) Follows from the multiplicativity of the determinant.
13. (b) Since $\operatorname{det} \phi$ is a homomorphism, its kernel is a normal subgroup of $G$. Since $G$ is simple, it must be that $\operatorname{ker}(\operatorname{det} \phi)={e}$ or $\operatorname{ker}(\operatorname{det} \phi)=G$. If the former is true, then det $\phi$ is injective, and its image is a subgroup of $\mathbb{C}^{\times}$, which is abelian. This $G$ would be isomorphic to an abelian group, but this cannot by by assumption. So it must be that $\operatorname{ker}(\operatorname{det} \phi)=G$, in which case $\operatorname{det} \phi$ is the trivial homomorphism.

Here’s a more complete solution:

(b) Let $\phi$ be a representation of $G$, and let $\chi_{\operatorname{det} \phi}$ denote the character of the one-dimensional representation given by $\operatorname{det} \phi$. Suppose $G$ is non-abelian and simple. Then $\ker(\chi_{\operatorname{det} \phi})$ is a normal subgroup of $G$. Since $G$ is non-abelian and simple, the only normal subgroups of $G$ are the trivial subgroup ${e}$ and $G$ itself. If $\ker(\chi_{\operatorname{det} \phi}) = G$, then $\chi_{\operatorname{det} \phi}$ is the trivial character. Otherwise, $\ker(\chi_{\operatorname{det} \phi}) = {e}$, so $\chi_{\operatorname{det} \phi}$ is injective. But the image of $\chi_{\operatorname{det} \phi}$ is a subgroup of $\mathbb{C}^\times$, which is abelian. Thus $G$ would be isomorphic to an abelian group, which is a contradiction. Therefore, $\ker(\chi_{\operatorname{det} \phi}) = G$, so $\chi_{\operatorname{det} \phi}$ is the trivial character.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

﻿