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This is the Bruhat order on $S_n / (S_k \times S_{n-k})$, which models the inclusion relations of Schubert varieties on the Grassmannian $Gr(k,n)$. The elements of $S_n / (S_k \times S_{n-k})$ are generally modeled by their minimal length representatives, called Grassmann permutations. You can biject your $k$-subset of $[n]$ to a Grassmann permutation by sending a $k$-subset $\lbrace\{ a_1, \dots, a_k \rbrace\}$ (with $a_1 < a_2 < \dots < a_k$) to the permutation of $[n]$ whose window (one-line) notation is $a_1 a_2 \cdots a_k b_1 b_2 \cdots b_{n-k}$ with $\lbrace\{ b_1, b_2, \dots, b_{n-k} \rbrace\}$ denoting the complement of your given set, again ordered in increasing order $b_1 < b_2 < \cdots < b_{n-k}$.

This is the Bruhat order on $S_n / (S_k \times S_{n-k})$, which models the inclusion relations of Schubert varieties on the Grassmannian $Gr(k,n)$. The elements of $S_n / (S_k \times S_{n-k})$ are generally modeled by their minimal length representatives, called Grassmann permutations. You can biject your $k$-subset of $[n]$ to a Grassmann permutation by sending a $k$-subset $\lbrace a_1, \dots, a_k \rbrace$ (with $a_1 < a_2 < \dots < a_k$) to the permutation of $[n]$ whose window (one-line) notation is $a_1 a_2 \cdots a_k b_1 b_2 \cdots b_{n-k}$ with $\lbrace b_1, b_2, \dots, b_{n-k} \rbrace$ denoting the complement of your given set, again ordered in increasing order $b_1 < b_2 < \cdots < b_{n-k}$.

This is the Bruhat order on $S_n / (S_k \times S_{n-k})$, which models the inclusion relations of Schubert varieties on the Grassmannian $Gr(k,n)$. The elements of $S_n / (S_k \times S_{n-k})$ are generally modelled by their minimal length representatives, called Grassmann permutations. You can biject your $k$-subset of $[n]$ to a Grassmann permutation by sending a $k$-subset consisting of $\{a_1, \dots, a_k\}$ (with $a_1 < a_2 < \dots < a_k$) to the permutation of $[n]$ whose window (one-line) notation is $a_1 a_2 \cdots a_k b_1 b_2 \cdots b_{n-k}$ with $\{b_1, b_2, \dots, b_{n-k}\}$ denoting the complement of your given set, again ordered in increasing order $b_1 < b_2 < \cdots < b_{n-k}$.

Edited because set braces don't seem to be displaying properly.

This is the Bruhat order on $S_n / (S_k \times S_{n-k})$, which models the inclusion relations of Schubert varieties on the Grassmannian $Gr(k,n)$. The elements of $S_n / (S_k \times S_{n-k})$ are generally modeled by their minimal length representatives, called Grassmann permutations. You can biject your $k$-subset of $[n]$ to a Grassmann permutation by sending a $k$-subset $\lbrace\{ a_1, \dots, a_k \rbrace\}$ (with $a_1 < a_2 < \dots < a_k$) to the permutation of $[n]$ whose window (one-line) notation is $a_1 a_2 \cdots a_k b_1 b_2 \cdots b_{n-k}$ with $\lbrace\{ b_1, b_2, \dots, b_{n-k} \rbrace\}$ denoting the complement of your given set, again ordered in increasing order $b_1 < b_2 < \cdots < b_{n-k}$.

This is the Bruhat order on $S_n / (S_k \times S_{n-k})$, which models the inclusion relations of Schubert varieties on the Grassmannian $Gr(k,n)$. The elements of $S_n / (S_k \times S_{n-k})$ are generally modelled by their minimal length representatives, called Grassmann permutations. You can biject your $k$-subset of $[n]$ to a Grassmann permutation by sending a $k$-subset consisting of $a_1, \dots, a_k$ (with $a_1 < a_2 < \dots < a_k$) to the permutation of $[n]$ whose window (one-line) notation is $a_1 a_2 \cdots a_k b_1 b_2 \cdots b_{n-k}$ with $b_1, b_2, \dots, b_{n-k}$ denoting the complement of your given set, again ordered in increasing order $b_1 < b_2 < \cdots < b_{n-k}$.

Edited because set braces don't seem to be displaying properly.

This is the Bruhat order on $S_n / (S_k \times S_{n-k})$, which models the inclusion relations of Schubert varieties on the Grassmannian $Gr(k,n)$. The elements of $S_n / (S_k \times S_{n-k})$ are generally modelled by their minimal length representatives, called Grassmann permutations. You can biject your $k$-subset of $[n]$ to a Grassmann permutation by sending a $k$-subset consisting of $\{a_1, \dots, a_k\}$ (with $a_1 < a_2 < \dots < a_k$) to the permutation of $[n]$ whose window (one-line) notation is $a_1 a_2 \cdots a_k b_1 b_2 \cdots b_{n-k}$ with $\{b_1, b_2, \dots, b_{n-k}\}$ denoting the complement of your given set, again ordered in increasing order $b_1 < b_2 < \cdots < b_{n-k}$.

Edited because set braces don't seem to be displaying properly.