9 problems

Little's conjecture. The two tableaux agree:

Symplectic Sagan–Worley recording-tableau conjecture. If $i>0$ and $a$ is any primed word, then

Orthogonal Knuth equivalence conjecture. Two primed words satisfy

Orthogonal Sagan–Worley insertion conjecture. For every $i\in\mathbb{Z}$,

Fixed-point-free involution bumping conjecture. For every $i\in[n]$, one has $\tilde w_i-w_i\in\{0,1,2\}$.

Involution-word bumping conjecture. For every $i\in[n]$, one has $\tilde w_i-w_i\in\{0,1\}$.

Let $v$ and $w$ be involution words, and let $\overset{\textsf{O}}\sim$ denote the shifted Hecke equivalence relation. The shifted Hecke insertion algorithm assigns to each word…

Let $v$ and $w$ be FPF-involution words, and let $\overset{\textsf{Sp}}\sim$ denote the symplectic Hecke equivalence relation. The symplectic Hecke insertion algorithm assigns to e…

Let an involution word be a word in the generators of the relevant involution-Hecke monoid, and let shifted Coxeter–Knuth equivalence be the equivalence relation generated by the f…