189-457B: Honors Algebra 4

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Lecture 1, on Monday January 6. Overview of the topics that will be covered. Definition of a linear representation of a finite group. Definition of an irreducible representation.

Lecture 2, on Wednesday January 8. A few basic examples of representations of finite groups. The special case of abelian groups. All irreducible representations of a finite abelian group are one-dimensional, and thus correspond to characters of the group. Representations of the symmetric group $S_3$ on three letters.

Lecture 3, on Friday January 10. Representations of the dihedral group $D_8$ of order $8$ and the quaternion group of order $8$. These groups admit a unique faithful irreducible complex representation; it is of dimension two.

Lecture 4, on Monday January 13. Semisimplicity of group representations and Maschke's theorem: 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 5, on Wednesday January 15. This lecture is posted on-line because I was out of town that day.
Proof of Maschke's theorem. The recorded lecture presents two proofs, the second involving the 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 $W$ of $V$, 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 6, on Friday January 17. This lecture formulates and discusses the theorem that, for any finite dimensional representation $\varrho$ of a finite group $G$, $$ \dim V^G = \frac{1}{\#G} \sum_{g\in G} {\rm Trace}(\varrho(g)),$$ where $V^G$ denotes the subspace of vectors in $V$ that are fixed by all the elements of $G$. When specialised to the case where $V = \mathbb C^X$ is the permutation representation attached to a $G$-set $X$, one recovers Burnside's counting Lemma. So the above formula for the dimension of $V^G$ can be envisaged as a generalisation of Burnside's formula, in the context of linear representations.

Lecture 7, on Monday January 20. Proof of the formula of Lecture 6, by observing that the linear endomorphism $$T:= \frac{1}{\#G} \sum_{g\in G} \varrho(g)$$ is a projection from $V$ onto $V^G$, and considering its trace. The general notion of characters (for non-abelian groups) as traces: given a representation $V$, the character $\chi_{_V}$ of $V$ is the function from $G$ to $\mathbb C$ defined by $$\chi_{_V}(g) := {\rm Trace}(\varrho_{_V}(g)), \quad \mbox{ for all } g\in G.$$ The notion of a character table, and a few examples (for $S_3$ and $D_8$.)

Lecture 8, on Wednesday January 22. Constructing new representations from existing ones. If $V_1$ and $V_2$ are two representations of $G$, then the vector space ${\rm hom}(V_1,V_2)$ is also a representation of $G$ by the rule $$ g * T := \varrho_{V_2}(g)\circ T \circ \varrho_{V_1}(g)^{-1}.$$ Its character is given by $$ \chi_{_{{\rm hom}(V_1,V_2)}}(g) = \overline{\chi_{_{V_1}}(g)} \chi_{_{V_2}}(g).$$ Furthermore, an element of ${\rm hom}(V_1,V_2)$ is $G$-invariant if and only if it is a homomorphism of $G$-representations.

Lecture 9, on Friday January 24. End of proof of the formula $$ \chi_{_{\hom(V,W)}}(g) = \overline{\chi_{_V}(g)}\chi_{_W}(g), \qquad \mbox{ for all } g\in G.$$ Corollary 1: the orthogonality of irreduclible characters: let $V_1, \ldots, V_t$ be irreducible representations and let $\chi_1,\ldots,\chi_t$ be their characters. Then $$ \langle \chi_i, \chi_j \rangle := \frac{1}{\#G} \sum_{g\in G} \overline{\chi_i(g)} \chi_j(g) = \left\{ \begin{array}{cl} 0 & \mbox{ if } i\ne j, \\ 1 & \mbox{ if } i = j. \end{array} \right.$$ Corollary 2: the irreducible characters are an orthonormal system of vectors in the space of class functions on $G$, relative to a natural hermitian pairing on this space. In particular, the number of non-isomorphic irreducible representations of $G$ is at most the number of conjugacy classes (class number) of $G$.

Lecture 10, on Monday January 27. The orthogonality of characters implies that if $\chi_V=\chi_W$, then $V$ and $W$ are isomorphic representations of $G$. I.e., a representation is determined up to isomorphism by its character.

Decomposition of the regular representation of $G$ into irreducible representations. Each irreducible representation occurs in the regular representation with multiplicity equal to its dimension. The sum of the squares of the dimensions of the irreducible representations of a group $G$ is equal to the cardinality of $G$.

The complex group ring $\mathbb C[G]$ is isomorphic to a direct sum of $d_i\times d_i$ matrix rings, where $d_1, \ldots,d_t$ are the dimensions of the distinct irreducible representations of $G$. Comparison of the centers of the group ring and the sum of matrix rings.
Corollary 1: the characters of irreducible representations of $G$ forms an orthonormal basis for the space of class functions on $G$.
Corollary 2: The number of distinct irreducible representations is equal to the class number (number of distinct conjugacy classes) of $G$.

Lecture 11, on Wednesday January 29. Introduction to fourier analysis on finite abelian groups.

Lecture 12, on Friday January 31. Fourier analysis on finite groups,continued. Application to the evaluation of sums of the form $$ \sum_{n=1}^\infty \frac{f(n)}{n},$$ where $f:\mathbb Z \rightarrow \mathbb C$ is a periodicfunction of period $N$.

Lecture 13, on Monday February 3. Fourier analysis on finite non-abelian groups. Application to random permutations and random walks on groups, developping a theme that was explored in the last few questions of Assignment 1.

Lecture 14, on Wednesday February 5. This lecture will be given on line.
The character tables of $S_4$ and $A_5$. Representations of dimensions $1$, $4$, and $5$ of $A_5$ are constructed using the standard permutation representations on $5$ letters and the transitive action of $A_5$ on $6$ letters. Construction of 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 15, on Friday February 7. This lecture will be given on line.
The character table of the simple group $G=GL_3(\mathbb Z/2\mathbb Z)$, of cardinality $168$. Three irreducible representations, of dimensions $1$, $6$, 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. To fill out one further line in this character table, we are motivated to introduce 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 $\psi$.

Lecture 16, on Monday February 10. Induced representations, continued. Formula for the character of an induced representation. If $V$ is the representation of $G$ induced from a character $\psi$ of a subgroup $H$, we showed that $$ \chi_V(g) = [G:H] \frac{1}{\#C(g)} \sum_{g\in H\cap C(g)} \psi(g),$$ where $C(g)$ is the conjugacy class of $g$.

Lecture 17, on Wednesday February 12. The formula for the character of an induced representation, continued.

Lecture 18, on Friday February 14. Application to the construction of an irreducible $8$-dimensional representation of the simple group of cardinality $168$. 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 two three-dimensional representations of $G$ cannot be conjugated into $GL_3(\mathbb R)$, i.e., they are not real. This remark provides the 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 Monday February 17. Tensor products of group representations. Review of the material and further topics/exercises.

Lecture 20, on Wednesday February 19. Applications of representation theory. Decomposition of the space of functions on the faces on a cube into irreducible representations.

Lecture 21, on Friday February 21. Review of representation theory.

Lecture 22, on Monday February 24. Review of representation theory.

Lecture 23, on Wednesday February 26. Midterm exam (in class) encompasing the material on representation theory that was covered in the first half of the class.

Lecture 24, on Friday February 28. Feedback from the midterm exam. Introduction of the second topic, Galois theory. Galois theory is a powerful tool in the study of fields and their extensions. Cardano's solution to the general cubic equation.

March 3-7: Study break.

Lecture 25, on Monday March 10 .

Field extensions. The degree of a field extension. The multiplicativity of the degree in towers of field extensions: if $F\subset K \subset E$ is a nested sequence of fields, then $$[F:E] = [F:K]\cdot [K:E].$$ (This is proved 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.

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. 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 26, on Wednesday March 12 . In this lecture we examined the fundamental notion of the automorphism group of a finite extension $E$ of $F$. It is the set of all field automorphisms of $E$ which are the identity on $F$. The main result we asserted is that the cardinality of $Aut(E/F)$ is always less than or equal to the degree of $E$ over $F$. When equality is attained, we say that the finite extension $E/F$ is Galois, and call $Aut(E/F)=Gal(E/F)$ the Galois group of $E/F$.

Lecture 27, on Friday March 14 . In this lecture we proved the fundamental fact about this group of symmetries of a finite extension $E$ of a field $F$: it is always finite, and its cardinality is bounded by the degree of $E/F$.

Lecture 28, on Monday March 17 .
This lecture was given on-line in recorded form.
The 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 $f(x)$. (I.e., it is the smallest extension of $F$ which contains all the roots of $f(x)$.) Abstract construction of the splitting field, by iteration of Kronecker's construction of the extension generated by a single root of an irreducible polynomial.

Lecture 29, on Wednesday March 19 .
This lecture was given on-line in recorded form.

The theory of finite fields. Construction of a field with $p^n$ elements, for $p$ a prime, as the splitting field of the polynomial $x^{p^n}-x$ in $F_p[x]$. The Galois group of $F_{p^n}$ over $F_p$ is a cyclic group of order $n$ with a canonical generator: the frobenius automorphism which sends $a\in F_{p^n}$ to $a^p$.

Lecture 30, on Friday March 21 .
Normality and seperability of a finite extension $E/F$. A finite extension is Galois if and only if it is normal and seperable.

Lecture 31, on Monday March 24 .
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 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 32, on Wedneday March 26 .

Lecture 33, on Friday March 28 .

Lecture 34, on Monday March 31 .

Lecture 35, on Wedneday April 2 .

Lecture 36, on Friday April 4 .

Lecture 37, on Monday April 7 .

Lecture 38, on Wedneday April 9 .

Lecture 39, on Friday April 11 .