21 problems

Schur positivity conjecture. The coefficients $a^\nu_{\lambda\mu}(t)$ are nonnegative integer polynomials in $t$.

Let $\lambda$ and $\mu$ be partitions, let $\widetilde{H}_{\mu}[X;0,t^{-1}]$ be the modified Hall–Littlewood polynomial specialized at $q=0$, let $\nabla$ be the nabla operator, an…

Let $\lambda$ and $\mu$ be partitions, let $\widetilde{H}_{\mu}[X;0,t]$ be the modified Hall–Littlewood polynomial obtained by specializing the modified Macdonald polynomial at…

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–…

The $Y=-1$ $h$-vector conjecture. The coefficients satisfy

Let $n$ and $k$ be the parameters in the Delta Conjecture, let $s_\lambda^\perp$ denote the operator adjoint to multiplication by $s_\lambda$, let $H_\lambda(x;q)$ be the Hall–Litt…

Let $Z\subset\mathbb{C}^n$ be any finite $S_n$-stable set such that $\mathbb{C}[x_1,\dots,x_n]/\operatorname{gr}\mathbf{I}(Z)$ is a graded $S_n$-module, where $\mathbf{I}(Z)$ is th…

Let $\Psi$ be a root ideal and let $mu$ be a partition. The Catalan function $H(\Psi;mu)$ is a symmetric function associated with this root ideal and partition. Chen–Haiman's conject…

Refined Hall–Littlewood filling identity conjecture.

Let $a$ and $b$ be positive integers, set $n=ab$, let $c$ be a divisor of $n$ with $a\leq c\leq b$, and set $d=n/c$. Write $\preceq_s$ for Schur positivity. The q-Doran conjecture.…

For $0<a\leq b$, let $F_{a,b}(\mathbf{x};q)$ be the divided difference from the $q$-Foulkes conjecture, and let the overline operator remove the largest part from each Schur index.…

For each positive integer $n$, let $H_n(\mathbf{x};q)$ be the Hall–Littlewood polynomial used in the paper, and for $0<a\leq b$ define … Let $f_{a,b}=h_b[h_a]-h_a[h_b]$. The q-Foul…

Let $\widetilde\Phi(z):\mathcal F_{\vec u}\to\mathcal F_{\vec v}$ be the crystal-limit vertex operator, with $\vec u=(u_1,u_2)$ and $\vec v=(v_1,v_2)$, and let…

Kerov's stochastic monotonicity conjecture. If $r<i$, then

Let $b4$ be a diagram with $n$ cells, let $f_b4$ be an injective function as in the injective Yamanouchi-word conjecture, and let $c0$ range over standard Yamanouchi words of parti…

Let $m,n\geq 1$. Let $h_{\lambda}^{(m)}(w,z;q)$ denote the modified Rogers–Szegő factor defined earlier, let $P'_{\lambda}$ be the modified Hall–Littlewood polynomial, and let…

Let $M=(M_1,\dots,M_m)\in\mathbb Z_{+}^m$ and set $m_0(\lambda):=\infty$. For a partition $\lambda$, write $\lambda'$ for its conjugate, $m_j(\lambda)$ for the multiplicity of the p…

Hall–Littlewood positivity conjecture. The coefficients $K_{\Lambda\Omega}(t)$ and $\bar K_{\Lambda\Omega}(t)$ defined by

Let $\lambda$ and $\mu$ be partitions with $\mu \subseteq \lambda$, and let $r\geq 0$. Write $e_r=P_{1^r}$. For a vertical strip $\alpha/\beta$, define … where…

Let $\lambda$ and $\mu$ be partitions with $\mu \subseteq \lambda$, and let $r\geq 0$. For a skew shape $\alpha/\beta$, write $\operatorname{sk}(\alpha/\beta)$ for the coefficient…