189-457B: Honors Algebra 4

Blog

Lecture 1, on Monday January 8. The first lecture was devoted to giving an overview of the topics that will be covered, and to defining the linear representations of a finite group.

Lecture 2, on Wednesday January 10. This lecture gave a few basic examples of representations of finite groups, and defined the notion of an irreducible representation.

Lecture 3, on Friday January 5. This recorded lecture, which aimed to make up for the lecture of January 5 which we missed, discusses the special case of abelian groups; notably, it is shown that all irreducible representations of a finite abelian group are one dimensional and thus correspond to characters of the group. It is also shows that the group of characters of a finite group $G$ is abstractly isomorphic to $G$.

Lecture 4, on Friday January 12. This Friday's lecture is also a recorded lecture since I am out of town that day. I discuss the basic set up for fourier analysis on finite group and give an application to the study of a certain random walk on the cyclic group of order N. My treatment contains a mistake and is only valid when N is odd - I let you ponder why! I was a bit rushed near the end and hope the explanations were not too hard to digest. (If they were... that is what office hours are for...)

Lecture 5, on Monday January 15. We discussed some examples of representations and irreducible representations, and proved, notably, that the quaternion group of order $8$ admits a unique faithful irreducible complex representation, which is of dimension two.

Lecture 6, on Wednesday January 17. We started by showing that the faithful irreducible representation of the quaternion group constructed in Lecture 5 cannot be conjugated into ${\bf GL}_2(\mathbb R)$, even though every matrix in the image is conjugate to a matrix with real entries. We then discussed the general process of linearisation. The lecture concluded with a discussion of semisimplicity of group representations and of Maschke's theorem.

Lecture 7, on Friday January 19. This lecture completed the proof of Maschke's theorem, which implies that every finite dimensional complex representation of a finite group $G$ is isomorphic to a direct sum of irreducible representations of $G$. The importance of this fact in the theory of group representations is hard to overstate.

Lecture 8, on Monday January 22. This lecture was devoted to an alternate proof of Maschke's theorem, which introduces the very useful notion of unitarisability: every finite dimensional vector space $V$ over $\mathbb C$ endowed with a linear action of a finite group $G$ can be equipped with a hermitian inner product, for which the group G acts by unitary transformations, i.e., for which $$ gv \cdot gw = v \cdot w, \qquad \mbox{ for all } g\in G, \ \ \ v,w\in V.$$ From this Maschke's theorem is immediate: given a subrepresentation $W$, the orthogonal complement $W'$ is the desired $G$-stable complementary subspace. The usefulness of the inner product structure for other purposes is also explained in the last questions in assignment 1, where it is used to derive information about the sizes of the eigenvalues of certain elements in the group ring acting on $V$.

Lecture 9, on Wednesday January 24. This lecture introduced the general notion of characters (for non-abelian groups) and prove their orthogonality. This implies that the number of non-isomorphic irreducible representations of $G$ is at most the number of conjugacy classes in $G$.

Lecture 10, on Friday January 26. This lecture showed that the sum of the squares of the dimensions of the irreducible representations of a group $G$ is equal to the cardinality of $G$, by analysing the decomposition of the regular representation of $G$ into irreducblbe representations.

Lecture 11, on Monday January 29. We showed that each irreducible representation occurs in the regular representation with multiplicity equal to its degree. It then follows that the complex group ring $\mathbb C[G]$ is isomorphic to a direct sum of $d_i\times d_i$ matrix rings, where the $d_i$ are the dimensions of the distinct irreducible representations of $G$. By comparing the centers of the group ring and the sum of matrix rings, it was finally shown that the number of irreducible representations is equal to the number of distinct conjugacy classes in $G$. In particular, the characters of irreudcible representations of $G$ forms an orthonormal basis for the space of class functions on $G$.

Lecture 12, on Wednesday January 31. The notion of a character table was introduced and the character tables of $D_8$ was described.

Lecture 13, on Friday February 2. Permutation representations were discussed and the relation between its character and fixed points was described.

Lecture 14, on Monday February 5. The character tables of $S_4$ and $A_5$ were described. Representations of dimensiona $1$, $4$, and $5$ of $A_5$ were constructed using the standard permutation representations on $5$ letters and the transitive action of $A_5$ on $6$ letters.

Lecture 15, on Wednesday February 7. We completed the character table of $A_5$ by constructing two distinct three-dimensional representations of $A_5$: the first arising from the group of rotations in $\mathbb R^3$ preserving the regular dodecahedron (or icosahedron), and the second obtained from the first by composing with an outer automorphism of $A_5$.

Lecture 16, on Friday February 9. We studied the character table of the group $G=GL_3(\mathbb Z/2\mathbb Z)$, a simple group of cardinality $168$. We constructed three irreducible representations, of dimensionas $1$, $6$, and $7$, arising from the permutation actions on the set of $7$ non-zero vectors in $(\mathbb Z/2\mathbb Z)^3$, and on the set of $8$ distinct Sylow $7$-subgroups, by conjugation.

Lecture 17, on Monday February 12. Motivated by the desire to construct an irreducible $8$-dimensional representation of the simple group of cardinality $168$, we introduced the notion of an induced representation of a group $G$. This construction allows one, starting with a subgroup $H$ of $G$ of index $t$ and a homomorphism $\psi: H \rightarrow \mathbb C^\times$, to build a $t$-dimensional representation of $G$, called the representation of $G$ induced from $\psi$. We almost gave a formula for the character attached to the induced representation.

Lecture 18, on Wednesday February 14. We completed the description of the character attached to an induced representation, and used this to calculate the $4$-th row of the character table of $G=GL_3(\mathbb Z/2\mathbb Z)$. Knowing that $G$ posseses $4$ irreducible representations, of dimensions $1$, $6$, $7$ and $8$, it directly follows that the remaining two irreducible representations of $G$ are of dimension three. Unlike the two irreducible three-dimensional representations of $A_5$, the three-dimensional representations of $G$ cannot be conjugated into $GL_3(\mathbb R)$, i.e., they are not real. This remark gives basis for calculating the characters of the two irreducible three-dimensional representations, thereby completing the character table for $G$, following some steps that are worked out in assignment $3$.

Lecture 19, on Friday February 16. Time permitting, I will discuss fourier analysis on finite non-abelian groups and its application to random walks on groups, developping a theme that was explored in the last few questions of assignment 1.

Lecture 20, on Monday February 19. We went over some questions in the previous assignment, notably the questions at the end of Assignment 1 related to how representation theory can be used to show the equidistribution of products of random elements in a finite group.

Lecture 21, on Wednesday February 21. More topics in representation theory, related to the character table of the simple group of order $168$.

Lecture 22, on Friday February 23. These lectures were devoted to various review topics in preparation for the midterm exam.

Lecture 23, on Monday February 26. These lectures were devoted to various review topics in preparation for the midterm exam.

Lecture 24, on Wednesday February 28. Midterm exam (in class) encompasing the material on representation theory that was covered in the first half of the class.

Lecture 25, on Friday March 1. In this final lecture before the spring (winter?) break, we embarked on our second topic, Galois theory. Galois theory is a powerful tool in the study of fields and their extensions, and so, after quickly recalling Cardano's solution to the general cubic equation, we began by discussing Kronecker's construction of field extensions.

Lecture 26, on Monday March 11 . We explained how the iteration of Kronecker's construction can be used to construct a splitting field of a polynomial $f(x)$ with coefficients in a field $F$. This splitting field is an extension $E\supset F$ over which $f(x)$ factors into linear factors, and which is generated as an extension of $F$ by all the roots of $F$. (I.e., $E$ is the smallest subfield of $E$ which contains both $F$ and all the roots of $f(x)$.)

After discussing the degree of a field extension of $F$, we then showed its main property, namely, its multiplicativity in sequences of field extensions: if $F\subset K \subset E$ is a nested sequence of fields, then $$[F:E] = [F:K]\cdot [K:E].$$ We proved this by showing that, if $a_1,\ldots, a_m$ is a basis for $E$ as a $K$-vector space, and $b_1,\ldots,b_n$ is a basis for $K$ as an $F$-vector space, then the collection of $mn$ elements of the form $a_i b_j$ form a basis for $E$ as an $F$-vector space.

The end of the lecture was devoted to the notion of constructibility. A constructible figure in the plane is a figure that can be obtained, starting from a collection of points with rational coordinates, by drawing lines through these points, circles with center at one of the points and containing another point, taking intersections of these figures and repeating as long as one wants. We mentioned that the angle bissector of two constructible lines intersecting in a point is also constructible, for example. A very popular problem (whose popularity persists to this day) was to find a similar procedure for trissecting an angle with ruler and compass. I indicated at the end of the lecture that the persistence of this problem in the popular imagination is largely undeserved, since it is in fact impossible to trissect a general (constructible) angle by ruler and compass. The proof of impossibility relies on nothing more than some basic analytic geometry (going back to the ideas of Descartes) and some basic field theory, involving nothing more than the multiplicativity of the degree in sequences of field extensions.

Lecture 27, on Wednesday March 13 . We explained how to prove the impossibility of effecting an angle trisection by ruler and compass. The first (key) step is to reduce this ostensibly geometric question to a question in field theory. Namely, a real number is said to be constructible if it can be obtained as a coordinate of a constructible point. We explained that, if $x$ is constructible, then is it is contained in a field $E_n$ that fits into a sequence $$ \mathbb Q = E_0 \subset E_1 \subset \cdots E_{n-1} \subset E_n,$$ where for each $0 \le j \le n-1$, the field $E_{j+1}$ is a quadratic extension of $E_j$, i.e, is of the form $E_{j+1} = E_j(\sqrt{a_j})$ for some $a_j\in E_j$.

This characterisation of constructible numbers puts a strong constraint on what they can be: they must all be contain in a finite extension of $\mathbb Q$ whose degree is a power of $2$! In particular, the root of an irreducible cubic polynomial with coefficients in $\mathbb Q$ cannot be constructible!

Lecture 28, on Friday March 15 . The question of the solvability of an equation by radicals can be recast in terms that are very similar to our study of constructible numbers: namely, a number can be expressed in terms of radicals (iterated extractions of $n$-th roots) if it is contained in a field $E_n$ that is obtained from a sequence $$ \mathbb Q = E_0 \subset E_1 \subset \cdots E_{n-1} \subset E_n,$$ where for each $0 \le j \le n-1$, the field $E_{j+1}$ is a radical extension of $E_j$, i.e, is of the form $E_{j+1} = E_j(\sqrt[m_j]{a_j})$ for some $a_j\in E_j$ and $m_j\ge 1$. Since the degree of $E_n$ is not restricted by this definition, it appears that the degree by itself is too crude an invariant to show that certain numbers cannot be expressed in terms of radicals.

The marvelous solution envisaged by Galois is to associate to an extension $E/F$ a more powerful and discerning invariant: the automorphism group $$ {\rm Aut}(E/F) = \{ f: E\rightarrow E \ \ \mbox{ s.t. } f(a+b) = f(a)+f(b), \ \ f(ab) = f(a) f(b), \ \ f(x) = x, \ \ \forall x\in F \}.$$ The study of this group and how it ineracts with the structure of the field is the main object of Galois theory. We thus will re-encounter, in a rather new setting, the theme that a lot of information about a mathematical object can be read off from its underlying group of symmetries.

Lecture 29, on Monday March 18 . A basic invariant of a finite extension $E/F$ of fields is its automorphism group Aut($E/F$), defines as the set of field isomorphisms from $E$ to itself that are the identity on $F$. In this lecture we proved a fundamental fact about this group of symmetries: it is always finite, and its cardinality is bounded by the degree of $E/F$.

Lecture 30, on Wednesday March 20 . A number of prototypical examples of pairs $E/F$ along with their automorphisms were examined. In particular, two examples of non-Galois extensions $E/F$ were given: the extension of the rationals generated by a cube root of $2$, and the extension of the field $F_p(t)$ of rational functions in an indeterminate $t$ over the field with $p$ elements, generated by a root of the polynomial $x^p-t$. These two examples give a reasonably complete list of what can prevent a finite extension $E/F$ from being Galois, i.e., from having as many automorphisms as its degree.

Lecture 31, on Friday March 22 .
This lecture explained what it means for a finite extension $E/F$ to be normal and seperable, and proved that the extension is Galois if and only if it is normal and seperable.