Problem 1.1 (yau-contest 2013)
Section: Algebra and Number Theory

Classify finite groups of order 26 up to isomorphisms.

Problem 1.2 (yau-contest 2013)
Section: Algebra and Number Theory

For each finite group $G$ of order $26$, describe the group $\operatorname{Aut}(G)$ of automorphisms of $G$.

Problem 2.1 (yau-contest 2013)
Section: Algebra and Number Theory

Consider $f \in \mathbb{Z}_{>0}$ and nonzero vector spaces $V_i$ indexed by $i \in \mathbb{Z}/f\mathbb{Z}$. Suppose that there are linear maps $\phi_i: V_i \to V_{i+1}$ and $\psi_i: V_i \to V_{i-1}$ such that
$\phi_{i-1} \circ \psi_i = 0$,\quad $\psi_{i+1} \circ \phi_i = 0$.
(We may think of a circular graph with oriented edges such that the “Orpheus condition” holds: Whenever you turn back while traveling through the graph you are killed.)
Prove that there exist lines $\ell_i \subset V_i$ for every $i \in \mathbb{Z}/f\mathbb{Z}$ such that $\phi_i(\ell_i) \subset \ell_{i+1}$, $\psi_i(\ell_i) \subset \ell_{i-1}$ under the condition: all $\psi_i = 0$.

Problem 2.2 (yau-contest 2013)
Section: Algebra and Number Theory

Consider $f \in \mathbb{Z}_{>0}$ and nonzero vector spaces $V_i$ indexed by $i \in \mathbb{Z}/f\mathbb{Z}$. Suppose that there are linear maps $\phi_i: V_i \to V_{i+1}$ and $\psi_i: V_i \to V_{i-1}$ such that
$\phi_{i-1} \circ \psi_i = 0$,\quad $\psi_{i+1} \circ \phi_i = 0$.
(We may think of a circular graph with oriented edges such that the “Orpheus condition” holds: Whenever you turn back while traveling through the graph you are killed.)
Prove that there exist lines $\ell_i \subset V_i$ for every $i \in \mathbb{Z}/f\mathbb{Z}$ such that $\phi_i(\ell_i) \subset \ell_{i+1}$, $\psi_i(\ell_i) \subset \ell_{i-1}$ under the condition: $\dim V_i$ are equal to each other.

Problem 3.1 (yau-contest 2013)
Section: Algebra and Number Theory

For a parameter $t = (t_0, t_1, \cdots, t_5) \in \mathbb{F}_5^6$ with $t_0 \ne 0$ and $\{t_i, i>0\}$ an ordering of elements in $\mathbb{F}_5$, define a polynomial
$$P_t(x) = (x - t_1)(x - t_2)(x - t_3) + t_0(x - t_4)(x - t_5).$$
Show that $P_t(x)$ is irreducible in $\mathbb{F}_5[x]$.

Problem 3.2 (yau-contest 2013)
Section: Algebra and Number Theory

For a parameter $t = (t_0, t_1, \cdots, t_5) \in \mathbb{F}_5^6$ with $t_0 \ne 0$ and $\{t_i, i>0\}$ an ordering of elements in $\mathbb{F}_5$, define
$$P_t(x) = (x - t_1)(x - t_2)(x - t_3) + t_0(x - t_4)(x - t_5).$$
Show that two parameters $t, t'$ give the same polynomial if and only if $t_0 = t_0'$ and $\{t_4, t_5\} = \{t_4', t_5'\}.$

Problem 3.3 (yau-contest 2013)
Section: Algebra and Number Theory

Show that every irreducible cubic monic polynomial over $\mathbb{F}_5$ is obtained by this way.

Problem 4.1 (yau-contest 2013)
Section: Algebra and Number Theory

For $k$ non-negative integer, let $V_k := \mathbb{R}[x]_{\le k}$ be the vector space of real polynomials of degree at most $k$ with an action by $\mathrm{SL}_2(\mathbb{R})$ by
$$\gamma \cdot P(x) = (cx + d)^k P\!\left(\frac{ax + b}{cx + d}\right), \quad \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{R}).$$
Show that $V_k$ is an irreducible representation of $\mathrm{SL}_2(\mathbb{R})$.

Problem 4.2 (yau-contest 2013)
Section: Algebra and Number Theory

For non-negative integers $m,n$, consider $V_{m,n} := V_m \otimes V_n$ as a subspace of $\mathbb{C}[x,y]$ of polynomials with both $x,y$-degrees at most $k$, with diagonal action of $\mathrm{SL}_2(\mathbb{R})$. Assume $m \ge n \ge 1$. Show that the following exact sequence is exact and split as representations of $\mathrm{SL}_2(\mathbb{R})$:
$$0 \longrightarrow V_{m-1,n-1} \xrightarrow{\langle y-x \rangle} V_{m,n} \xrightarrow{y=x} V_{m+n} \longrightarrow 0.$$
This implies the following decomposition of representations:
$$V_m \otimes V_n = \bigoplus_{i=0}^{n} V_{m+n-2i}.$$

Problem 4.3 (yau-contest 2013)
Section: Algebra and Number Theory

For non-negative integers $\ell \ge m \ge n$ consider the space of invariants $(V_\ell \otimes V_m \otimes V_n)^{\mathrm{SL}_2(\mathbb{R})}$. Show that this space is either trivial or one-dimensional; it is non-trivial if and only if
$$\ell + m + n \equiv 0 \pmod{2}, \qquad \ell + m \ge n.$$

Problem 5.1 (yau-contest 2013)
Section: Algebra and Number Theory

Find a polynomial $f(x)$ with integer coefficients which has a root over $\mathbb{F}_p$ for each prime $p$ but has not root over $\mathbb{Q}$.

Problem 5.2 (yau-contest 2013)
Section: Algebra and Number Theory

Can you find $f$ irreducible?

Problem 5.3 (yau-contest 2013)
Section: Algebra and Number Theory

What is the smallest possible degree of $f$?