39 problems

Let $u,v\in S_n$ be vexillary permutations, let $\phi(w)$ be the Wachs flag of $w$, and write $a\preceq b$ when $a$ is a prefix of $b$. Let $\mathfrak S_w(\mathbf x;\mathbf y)$ be…

Let $B$ be a $k$-row reduced bumpless pipe dream such that row $k$ has no blanks (“weighty tiles”). Let $\phi(B)$ be the RC graph corresponding to $B$ under the natural bijection b…

Let $(\mathbf n,\mathbf m)$ be as above, and let $\boldsymbol\alpha_{\mathbf n}$ and $\boldsymbol\beta_{\mathbf m}$ be positive sequences satisfying the stated conditions and summa…

Layered exponential-growth conjecture. The two maxima have the same exponential growth rate:

Convergence-to-the-Schubert-permuton conjecture. The random permutations distributed according to this measure converge, as $n\to\infty$, to a deterministic permuton $\mu$ on…

Let $w$ be a permutation, let $\mathfrak{S}_w$ denote its Schubert polynomial, and consider its expansion in the SEM basis. A polynomial is a single SEM if it is one basis element…

Key-polynomial operator conjecture. For any composition $\alpha$,

Down-up Monk connectedness conjecture. For any $u,v\in S_{\mathbb Z}$, the set $\\{w\mid\overleftarrow{c_{u,v}^w}\ne0\\}$ is down-up Monk connected.

Commuting-map conjecture. There exists a map $\psi$ satisfying Proposition such that

For $w\in S_n$, write … where $\theta_w$ is the number of supports of $\mathfrak{S}_w(x)$ and $p_u(w)$ counts occurrences of the permutation pattern $u$ in $w$. The coefficients ar…

For $w\in S_n$, write … where $p_u(w)$ counts occurrences of the pattern $u$ in $w$, and define the coefficients recursively by … Here $u$ ranges over permutations and $c_u$ has th…

For $n\geq 1$, let $\theta_w$ denote the number of supports of the Schubert polynomial $\mathfrak{S}_w(x)$, equivalently the number of lattice points in its Newton polytope, and de…

For $n\geq 1$, define … where $\nu_w=\mathfrak{S}_w(x)|_{x_i=1}$. A layered permutation $w(b_1,\ldots,b_k)$ is the concatenation of decreasing blocks on consecutive intervals, wher…

Let $S_n$ be the symmetric group on $n$ letters. For permutations $u,w\in S_n$, let $D_u^w$ denote the dual Schubert polynomial, and let SCNP denote the property that its Newton po…

Let $cmathfrak{S}_{w}cin coperatorname{Pol}_n$ be a Schubert polynomial, and let $clangle cmathfrak{S}_{w}crangle_n^q$ denote its $q$-divided symmetrization. For a partition…

Let $cmathfrak{S}_{w}(x_1,\cdots,x_n)$ be a Schubert polynomial, and let $clangle cmathfrak{S}_{w}crangle_n^q$ denote its $q$-divided symmetrization. A symmetric polynomial is Hall–…

Let $D$ be a diagram, and write its flagged Schur character in the Schubert-polynomial basis as … where the sum is over all permutations and $c^w_D$ are integers. A diagram is call…

Let $w\in S_n$, let $\mathfrak{S}_w(x)$ be the Schubert polynomial indexed by $w$, and let $N$ be the normalization operator on monomials, defined by … A homogeneous polynomial is…

Fix $k\in\mathbb{Z}_{>0}$, let $\zeta$ be a $k$-th root of unity, and let $\Phi_w^k$ denote the principal specialization $\mathfrak{S}_w(1,q,q^2,\ldots)$ at $q=\zeta$. Define … and…

Doubly layered maximizer conjecture. For every $n$, all permutations reaching the maximum $\widetilde{u}(n)$ are doubly layered permutations. Thus, there is a limit

Let $w\in S_n$ be a vexillary permutation. Write $\mathfrak G_w^\mathrm{top}$ for the top-degree homogeneous component of its Grothendieck polynomial, and let $D^\mathrm{top}(w)$ b…

Let $\mathcal{E}_n$ be the Fomin–Kirillov algebra generated by $\mathbf{x}_{ab}$ for $1\leq a<b\leq n$, and define the commuting Dunkl elements … Let $\mathfrak{S}_w$ denote the Sc…

Fix $w\in S_n$ and any term order with $x_1>x_2>\cdots>x_n$. Let $\mathfrak{G}_w^{(k)}$ denote the degree-$k$ homogeneous component of $\mathfrak{G}_w$, and let $m_k(\bm{x})$ be it…

Fix $w\in S_n$. A heavy climbing chain of $w$ is a climbing chain together with its marked-link data $M(C)$; write its overlined and ordinary weights as…

Multiplicity-free inverse Grassmannian conjecture. If $u$ and $v$ are inverse Grassmannian, then $\mathfrak S_u\cdot\mathfrak S_v$ is a multiplicity-free sum of Schubert polynomial…