Groups and their Representations

Lecture 17

Homework

1. Assume that $k\leq 2n$. Let $$ V_k = \Big\{ f: \binom nk\to \QQ\mid \sum_{\{s\in \binom nk\mid s\supset t\}} f(s) = 0 \text{ for all } t\in \binom n{k-1}\Big\}. $$ Compute the dimension of $V_k$.
2. Show that $V_k$ is an irreducible representation of $S_n$.

Lecture 16

Homework

Lecture 15

Homework

1. Recall that the algebra $\CC[\binom n2]$ has basis $T_i:=T_{k_i}$ with $i = 0, 1, 2$. Compute the products $T_iT_j$ in this algebra. Find its primitive idempotents.

Lecture 14

Lecture 13

Homework

1. Let $\rho$ be the two-dimensional representation of the cyclic group $C_n$ defined by $\rho(1) = \begin{pmatrix}\cos(2\pi/n) & \sin(2\pi/n) \\ -\sin 2\pi/n & \cos 2\pi/n\end{pmatrix}$. Show that $\mathrm{End}_{C_4}\QQ^2$ is the field obtained by adjoining a primitive $n$th root of unity to $\QQ$.
2. Let $f$ be a polynomial with distinct roots. Let $A$ be a matrix with $f(A) = 0$. Show that $A$ is diagonalizable.
3. Determine whether or not the converse of Schur's lemma is true (assume $V$ finite dimensional over an algebraically closed field; does there exist a representation such that $\mathrm{End}_G V = K$, but $V$ is not simple.

Lecture 12

Homework

1. The GAP database of transitive groups shows two groups isomorphic to $S_5$ of degree $10$ (nos. 12 and 13). Which of them is conjugate in $S_{10}$ to the subgroup from Lecture 11?
2. Consider the action of $S_5$ on $\binom 52 \times \binom 52$ given by $w\cdot(x, y) = (w\cdot x, w\cdot y)$. How many orbits does this action have? Also, how many orbits does the the restriction of this action to $A_5$ have?
3. List representatives of the conjugacy classes of $A_5$ along with their cardinalities.

Lecture 11

Homework

1. Let $V$ denote the space of all functions on the set of vertices of the regular dodecahedron such that
   1. the values add up to zero on each face
   2. the value at each vertex is equal to the value at its antipode.
   With these definitions,
   1. compute the dimension of $V$.
   2. show that $V$, as a representation of the group of rigid motions which map the vertex-set of the dedecahedron onto itself (we know that this group is isomorphic to $A_5$), is irreducible.

Lecture 10

Homweork

1. Let $V_n$ be the $n-1$-dimensional space of vectors in $\QQ^n$ whose coordinates add up to $0$. The symmetric group $S_n$ acts on this space by permuting the coordinates of these vectors: $$ r_n(w)(x_1,\dotsc, x_n) = (x_{w(1)},\dotsc, x_{w(n)}). $$ View $S_{n-1}$ as the subgroup of $S_n$ which permutes the elements $1,\dots,n-1$ of $\{1,\dotsc, n\}$. Show that the restriction of $(r_n,V_n)$ is the sum of the trivial representation of $S_{n-1}$ and the representation $(r_{n-1}, V_{n-1})$.
2. Show that $(r_n, V_n)$ is an irreducible representation of $S_n$ for each positive integer $n$.
3. Show that $(r_n, V_n)$ is not isomorphic to $(r_n\otimes \epsilon, V_n)$ for any $n\geq 2$.

Lecture 9

Lecture 8

Homework

1. Let $V$ be the set of functions on the vertices of an octahedron whose sum, over any set of vertices which form a triangular face is $0$. Choose two adjecent vertices and call them $1$ and $2$. Let $v_1$ denote the unique element of $V$ whose value at $1$ is $1$ and value at $2$ is $0$. Likewise, let $v_2$ denote the unique element of $V$ whose value at $1$ is $1$ and value at $1$ is $0$. Then $v_1$ and $v_2$ form a basis of $V$. Write down the matrices for the action of any set of generators $S_4$ (the group of rigid motions that fixes the set of vertices of the octahedron). Conlude that $V$ is a simple representation.
2. With $T$ as above, and $\rho:S_4\to GL(V)$ denoting the action of $S_4$ on it, find a linear isomorphism $T:V\to V$ such that $$\rho(w)T = \epsilon(w)T\rho(w).$$
3. Let $V$ be the space of all functions from the set of vertices of an icosahedron to $\QQ$, the sum of whose values on each face is $0$. What is $\dim V$? Note: You may want to take this into account before you shout out your answer.

Lecture 7

Homework

1. Show that the action of the group of rigid motions of of the dodecahedron on its set of five embedded cubes defines an isomorphism from this group onto $A_5$.
2. Consider the representation of $S_4$ on the three-dimensional subspace $V$ of vectors in $\mathbf Q^4$ whose coordinates add up to $0$. Is there proper subspace of $V$ that is mapped to itself (i.e., is invariant) under this action?

Lecture 6

Homework

1. Show that the automorphism group of the graph on the top of the first page of the notes is isomorphic to the dihedral group with eight elements.
2. Decide whether or not the automorphism group of the cube graph is isomorphic to $S_4\times C_2$ (here $C_2$ is the cyclic group with two elements).

Lecture 5

Homework

1. Let $G$ be the group of symmetries of the cube. Let $\Gamma$ be the graph whose vertices are the corners of the cube, and edges are the edges of the cube. Let $\mathrm{Aut}(\Gamma)$ denote the group of bijections of the vertices of $\Gamma$ which preserve edge relations. What is the order of this group? What is the relationship between this group and $G$?
2. Repeat the previous exercise with the cube replaced by the tetrahedron, and by the dodecahedron.
3. Show that the group of rigid motions that preserve the dodecahedron is isomorphic to $A_5$. Click here for a hint; also try using the dodecahedral graph to make your work easier.

Lecture 4

Homework

1. Consider the following algorithm:
   1. Start with the permutation $w = 12\dotsb n$.
   2. As $i$ goes from $1$ to $n$ do the following:
      • swap the element $w_i$ with a random element from $w_{i+1},\dotsc, w_n$.
   This is a minor change from Problem 3 from Lecture 1 (there $w_i$ was swapped with a random element from $w_i, \dotsc, w_m$). What is the resulting probability distribution on $S_n$?
2. Let $\langle \vec x, \vec y\rangle$ denote the dot product of vectors in $\RR^3$, and $\|\vec x\|^2 = \langle \vec x, \vec x \rangle$. Show that if $\|A \vec v\| = \|\vec v\|$ for each $\vec v \in \RR^3$, then $\langle A\vec v, A\vec w\rangle = \langle \vec v, \vec w\rangle$ for all $\vec v, \vec w\in \RR^3$.
3. We have seen that $G_{(0,0,0)}$ the group of rigid motions of space which fix the origin is $SO(3)$, the group of orthogonal matrices with determinant one. Find a nice description of the group $G_{(1,0,0)}$, the group of rigid motions which fix the first coordinate vector.
4. Think of $\RR^3$ as the subspace of $\RR^4$ consisting of vectors whose coordinates add up to $1$. In this space, the coordinate vectors $(1,0,0,0)$, $(0, 1, 0, 0)$, $(0, 0, 1, 0)$ and $(0, 0, 0, 1)$ form the vertices of a regular tetrahedron. Use this idea to write down the coordinates of four points in $\RR^3$ which form the corners of a regular tetrahedron.
5. Express the element $(12)(34)$ of $S_4$ as a composition of $3$-cycles.
6. Does there exist a configuration $C$ of four points in $\RR^3$ such that the group of rigid motions which fix $C$ has $24$ elements?
7. Is the group of rigid motions that fix the set of vertices of a cube isomorphic to the symmetric group? Note: the vertices of a cube can be taken to be the eight vectors of the form $(\pm 1, \pm 1, \pm 1)$.
8. Is the group of rigid motions that fix the set of vertices of a cube isomorphic to the group of rigid motions of a regular octahedron? Note: the vertices of a regular octahedron can be taken to be the three coordinate vectors in $\RR^3$ and their negatives.

Lecture 3

Homework

1. Find the Knuth shuffle decomposition of the permutation $w = 7653124$, i.e., express this permutation as a product $$ (i_1 j_1)(i_2 j_2)\dotsb (i_k j_k), $$ where $1\leq i_1< i_2 \dotsb < i_k\leq n$, and $i_r< j_r$ for each $1\leq r\leq k$ (here $n= 7$).
2. Find an algorithm to write any permutation as its Knuth shuffle decomposition (see pervious question for definition of Knuth shuffle decomposition).

Lecture 2

Homework

1. Let $w\in S_n$ be the permutation for which $w(i) = j$ and $w(j) = i$ for some $1\leq i < j \leq n$. Compute $\mathrm{inv}(w)$.
2. Recall that a polynomial $a_0 + a_1t + \dotsb + a_Nt^N$ with positive coefficients of degree $N$ is called symmetric unimodal if $a_i = N_{N-i}$ for each $0\leq i\leq N$, and $a_i\leq a_{i+1}$ for $0\leq i \lt N/2$. Show that a product of symmetric unimodal polynomials is symmetric unimodal. Conculde that the distribution of the inversion number statistic on $S_n$ is unimodal.
3. The rank of a permutation $w\in S_n$ is defined to be its position in the list of all permutations written out in lexicographic order. For example, $231$ has rank $4$. Compute the rank of the permutation $64312875$ without using a computer. Check your answer with Sage.
4. How many permutations of $8$ have cycle type $(3, 2, 2, 1)$?

Lecture 1

Homework

1. Suppose that $\alpha, \beta: 2^{\mathbf n}\to \mathbf C$ satisfy the identity: $$ \alpha(I) = \sum_{J\supset I} (-1)^{|J|-|I|}\beta(J) \text{ for all } I, J \subset \mathbf n. $$ Show that $$ \beta(I) = \sum_{J\supset I} \alpha(J) \text{ for all } I, J \subset \mathbf n. $$
2. Suppose that a vector $\mathbf x \in [0, 1]^n$ is chosen at random, and then the coordinates of this vector, taken in increasing order are replaced by the integers $1,2,\dotsc, n$, then the probability that the result is a given permutation $w$ of size $n$ is $1/n!$.
3. Consider the following algorithm:
   1. Start with the permutation $w = 12\dotsb n$.
   2. As $i$ goes from $1$ to $n$ do the following:
      • swap the element $w_i$ with a random element from $w_i,\dotsc, w_n$.
   Show that the probability of ending up with any given permutation is $1/n!$.