题号:14618    题型:解答题    来源:2024年丘成桐大学生数学竞赛(代数与数论类)-无答案
Let $n \geq 1$ be an integer. Let $A$ be a discrete valuation ring wit h $K$ its field of fractions and $\pi \in A$ a uniformizer. For $\lambda=\left(\lambda_1, \cdots, \lambda_n\right) \in \mathbb{Z}^n$ write
$$
D_\lambda=\operatorname{diag}\left(\pi^{\lambda_1}, \cdots, \pi^{\lambda_n}\right)=\left(\begin{array}{cccc}
\pi^{\lambda_1} & 0 & \cdots & 0 \\
0 & \pi^{\lambda_2} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \pi^{\lambda_n}
\end{array}\right) \in \mathrm{GL}_n(K) .
$$

Show that, for $\lambda, \mu \in \mathbb{Z}^n$, the following intersection inside $\mathrm{GL}_n(K)$
$$
\mathrm{GL}_n(A) \cdot D_\mu \cdot \mathrm{GL}_n(A) \bigcap U(K) \cdot D_\lambda
$$
is non-empty if and only if $\lambda_{\text {dom }} \leq \mu_{\text {dom }}$. Here
- $\mathrm{GL}_n(K)$ (resp. $\mathrm{GL}_n(A)$ ) is the group of invertible $n \times n \mathrm{sq}$ uare matrices with coefficients in $K$ (resp.in $A$ ), and $U(K) \subset \mathrm{GL}_n(K)$ is the standard unipotent subgroup, that $\mathrm{i}$ $\mathrm{s}$, the subgroup of upper triangular matrices with coefficients 1 on the diagonal.
- for $\alpha=\left(a_1, \cdots, a_n\right)$ and $\beta=\left(b_1, \cdots, b_n\right)$ two elements $\mathrm{i}$ $\mathrm{n} \mathbb{Z}^n$, we write $\alpha \leq \beta$ if
$$
\sum_{i=1}^k a_i \leq \sum_{i=1}^k b_i, \quad \text { for any } 1 \leq k \leq n
$$
and if $\sum_{i=1}^n a_i=\sum_{i=1}^n b_i$. Write also $\alpha_{\text {dom }}:=\left(a_1^{\prime}, \cdots, a_n^{\prime}\right)$ with $a_1^{\prime}, \cdots, a_n^{\prime}$ an arrangement of $a_1, \cdots, a_n$ such that
$$
a_1^{\prime} \geq a_2^{\prime} \geq \cdots \geq a_n^{\prime}
$$
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