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Let $Q$ be a word in the generators of some Coxeter group, and consider a subword $R$ (not necessarily reduced, though I might want $Q$ to be).

Define the greedy or Demazure product of $R$ as follows: while multiplying generators left-to-right, insist on only going upward in Bruhat order; any letter in $R$ that would take you downward should be skipped over.

For abstruse geometric reasons, I want to divide the set of all $2^{|Q|}$ subwords of $Q$ into equivalence classes $\{R\}$, according to (the greedy product of $R$, the locations in $Q$ of the skipped letters). That is, for each pair $(w \in W, S\subseteq Q)$, I consider the set of $R$ with greedy product $w$ and skip locations at exactly $S$.

For example, each $(w,\emptyset)$ corresponds to the class of reduced subwords with (ordinary) product $w$, a well-studied object (e.g. in my paper with Ezra Miller on subword complexes). If $Q=121$ in $S_3$, the $2^3$ subwords break into a singleton for each $w$, except for $\{1--,--1\}$ and $\{1-\underline{1}\}$ for $w=(12)$, where the underlined $\underline{1}$ is the only skipped letter.

Has anyone studied these equivalence classes of subwords before?

 

If so, are standard results like the Coxeter exchange condition known for them?

EDIT: here's a larger example, hopefully to make clearer the equivalence relation. Let $Q = 1231$, a word in the generators of $S_4$. Then the classes are $$ ---- $$ $$ 1---, \quad ---1 $$ $$ -2-- $$ $$ --3- $$ $$ 12-- $$ $$ 1-3-, \quad --31 $$ $$ 1--\underline{1} $$ $$ -23- $$ $$ -2-1 $$ $$ 123- $$ $$ 12-1 $$ $$ 1-3\underline{1} $$ $$ -231 $$ $$ 1231 $$ where I've underlined the skip positions.

Let $Q$ be a word in the generators of some Coxeter group, and consider a subword $R$ (not necessarily reduced, though I might want $Q$ to be).

Define the greedy or Demazure product of $R$ as follows: while multiplying generators left-to-right, insist on only going upward in Bruhat order; any letter in $R$ that would take you downward should be skipped over.

For abstruse geometric reasons, I want to divide the set of all $2^{|Q|}$ subwords of $Q$ into equivalence classes $\{R\}$, according to (the greedy product of $R$, the locations in $Q$ of the skipped letters). That is, for each pair $(w \in W, S\subseteq Q)$, I consider the set of $R$ with greedy product $w$ and skip locations at exactly $S$.

For example, each $(w,\emptyset)$ corresponds to the class of reduced subwords with (ordinary) product $w$, a well-studied object (e.g. in my paper with Ezra Miller on subword complexes). If $Q=121$ in $S_3$, the $2^3$ subwords break into a singleton for each $w$, except for $\{1--,--1\}$ and $\{1-\underline{1}\}$ for $w=(12)$, where the underlined $\underline{1}$ is the only skipped letter.

Has anyone studied these equivalence classes of subwords before?

If so, are standard results like the Coxeter exchange condition known for them?

EDIT: here's a larger example, hopefully to make clearer the equivalence relation. Let $Q = 1231$, a word in the generators of $S_4$. Then the classes are $$ ---- $$ $$ 1---, \quad ---1 $$ $$ -2-- $$ $$ --3- $$ $$ 12-- $$ $$ 1-3-, \quad --31 $$ $$ 1--\underline{1} $$ $$ -23- $$ $$ -2-1 $$ $$ 123- $$ $$ 12-1 $$ $$ 1-3\underline{1} $$ $$ -231 $$ $$ 1231 $$ where I've underlined the skip positions.

Let $Q$ be a word in the generators of some Coxeter group, and consider a subword $R$ (not necessarily reduced, though I might want $Q$ to be).

Define the greedy or Demazure product of $R$ as follows: while multiplying generators left-to-right, insist on only going upward in Bruhat order; any letter in $R$ that would take you downward should be skipped over.

For abstruse geometric reasons, I want to divide the set of all $2^{|Q|}$ subwords of $Q$ into equivalence classes $\{R\}$, according to (the greedy product of $R$, the locations in $Q$ of the skipped letters).

For example, each $(w,\emptyset)$ corresponds to the class of reduced subwords with (ordinary) product $w$, a well-studied object (e.g. in my paper with Ezra Miller on subword complexes). If $Q=121$ in $S_3$, the $2^3$ subwords break into a singleton for each $w$, except for $\{1--,--1\}$ and $\{1-\underline{1}\}$ for $w=(12)$, where the underlined $\underline{1}$ is the only skipped letter.

Has anyone studied these equivalence classes of subwords before?

If so, are standard results like the Coxeter exchange condition known for them?

Let $Q$ be a word in the generators of some Coxeter group, and consider a subword $R$ (not necessarily reduced, though I might want $Q$ to be).

Define the greedy or Demazure product of $R$ as follows: while multiplying generators left-to-right, insist on only going upward in Bruhat order; any letter in $R$ that would take you downward should be skipped over.

For abstruse geometric reasons, I want to divide the set of all $2^{|Q|}$ subwords of $Q$ into equivalence classes $\{R\}$, according to (the greedy product of $R$, the locations in $Q$ of the skipped letters). That is, for each pair $(w \in W, S\subseteq Q)$, I consider the set of $R$ with greedy product $w$ and skip locations at exactly $S$.

For example, each $(w,\emptyset)$ corresponds to the class of reduced subwords with (ordinary) product $w$, a well-studied object (e.g. in my paper with Ezra Miller on subword complexes). If $Q=121$ in $S_3$, the $2^3$ subwords break into a singleton for each $w$, except for $\{1--,--1\}$ and $\{1-\underline{1}\}$ for $w=(12)$, where the underlined $\underline{1}$ is the only skipped letter.

Has anyone studied these equivalence classes of subwords before?

If so, are standard results like the Coxeter exchange condition known for them?

EDIT: here's a larger example, hopefully to make clearer the equivalence relation. Let $Q = 1231$, a word in the generators of $S_4$. Then the classes are $$ ---- $$ $$ 1---, \quad ---1 $$ $$ -2-- $$ $$ --3- $$ $$ 12-- $$ $$ 1-3-, \quad --31 $$ $$ 1--\underline{1} $$ $$ -23- $$ $$ -2-1 $$ $$ 123- $$ $$ 12-1 $$ $$ 1-3\underline{1} $$ $$ -231 $$ $$ 1231 $$ where I've underlined the skip positions.

Let $Q$ be a word in the generators of some Coxeter group, and consider a subword $R$ (not necessarily reduced).

Define the greedy or Demazure product of $R$ as follows: while multiplying generators left-to-right, insist on only going upward in Bruhat order; any letter in $R$ that would take you downward should be skipped over.

For abstruse geometric reasons, I want to divide the set of all $2^{|Q|}$ subwords of $Q$ into equivalence classes $\{R\}$, according to (the greedy product of $R$, the locations in $Q$ of the skipped letters).

For example, each $(w,\emptyset)$ corresponds to the class of reduced subwords with (ordinary) product $w$, a well-studied object (e.g. in my paper with Ezra Miller on subword complexes). If $Q=121$ in $S_3$, the $2^3$ subwords break into a singleton for each $w$, except for $\{1--,--1\}$ and $\{1-\underline{1}\}$ for $w=(12)$, where the underlined $\underline{1}$ is the only skipped letter.

Has anyone studied these equivalence classes of subwords before?

If so, are standard results like the Coxeter exchange condition known for them?

Let $Q$ be a word in the generators of some Coxeter group, and consider a subword $R$ (not necessarily reduced, though I might want $Q$ to be).

Define the greedy or Demazure product of $R$ as follows: while multiplying generators left-to-right, insist on only going upward in Bruhat order; any letter in $R$ that would take you downward should be skipped over.

For abstruse geometric reasons, I want to divide the set of all $2^{|Q|}$ subwords of $Q$ into equivalence classes $\{R\}$, according to (the greedy product of $R$, the locations in $Q$ of the skipped letters).

For example, each $(w,\emptyset)$ corresponds to the class of reduced subwords with (ordinary) product $w$, a well-studied object (e.g. in my paper with Ezra Miller on subword complexes). If $Q=121$ in $S_3$, the $2^3$ subwords break into a singleton for each $w$, except for $\{1--,--1\}$ and $\{1-\underline{1}\}$ for $w=(12)$, where the underlined $\underline{1}$ is the only skipped letter.

Has anyone studied these equivalence classes of subwords before?

If so, are standard results like the Coxeter exchange condition known for them?