595 problems

For $n=2$, write the Chern class in the Schur basis as … and expand each coefficient in the binomial basis: … A polynomial is binomially positive when all its binomial-basis coeffi…

Generalized Amdeberhan–Shareshian–Stanley conjecture. For every partition $\mu$ of $kn$, the symmetric function obtained by applying the partition division map…

Let $n$ be a positive integer, let $\lambda\vdash n$ be a partition, and let $\phi^\lambda$ denote the corresponding monomial trace. For an $n\times n$ matrix $x$, write…

Let $n\geq 1$. For partitions $\pi,\mu,\nu\vdash n$, define the coefficient $f^\pi_{\mu,\nu}(\alpha)$ by … where $e_\pi$ denotes the elementary symmetric function. Monomial positiv…

Let $R_n^{(1,3)}$ be the $(1,3)$-bosonic-fermionic coinvariant ring, let $\operatorname{Frob}(R_n^{(1,3)};1;1,1,1)$ denote its Frobenius series with all grading variables specializ…

Let $R_n^{(1,2)}$ be the $(1,2)$-bosonic-fermionic coinvariant ring, and let $\operatorname{Frob}(R_n^{(1,2)};q;u,v)$ denote its multigraded Frobenius series. The notation…

Let $R_n^{(2,2)}$ be the $(2,2)$-bosonic-fermionic coinvariant ring, and let $\operatorname{Frob}(R_n^{(2,2)};q,t;u,v)$ denote its multigraded Frobenius series. Here $\Theta_{e_k}$…

Let $I_\infty$ be the involutions of the infinite symmetric group, let $\mathrm{Des}_V(z)$ be the visible descent set, and let $\mathfrak G^{\mathsf{O}}_z$ be the orthogonal involu…

Let $\alpha$ and $\gamma$ be weak compositions, with $\alpha$ skew-symmetric. Say that $\kappa^\mathsf{P}_\alpha-\kappa_\gamma$ is key positive when it is a nonnegative linear comb…

Let $\alpha$ be a symmetric weak composition, and let a P-key polynomial mean a polynomial of the form $\kappa^\mathsf{P}_\eta$ for a skew-symmetric weak composition $\eta$. Conver…

Let $I_\infty$ denote the involutions of the infinite symmetric group, let $\mathfrak S^{\mathsf{O}}_z$ be the corresponding orthogonal involution Schubert polynomial, and let…

Let $\alpha$ be a skew-symmetric weak composition, let $n=\ell(\alpha)$, and let $\widetilde\rho(\alpha)$ and $\widetilde\gamma(\alpha)$ denote its strict sub-diagonal row and colu…

Let $\alpha$ and $\gamma$ be distinct symmetric weak compositions. Injectivity conjecture. Then … This conjecture is supported by computations, although the authors report no heuri…

Let $u_r$ be the symmetric homogeneous polynomial coordinates in $N$ variables constructed in the main theorem. For a partition $\sigma\vdash r$ and a differential operator of orde…

Codominant-support conjecture. The local system $L_{z,w}^{J',\emptyset}$ is zero for every non-codominant permutation $z$. The conjecture would restrict the decomposition-theorem s…

Parabolic h-positivity conjecture. The Frobenius character

Permutation-representation conjecture. The local systems $L_{z,w}^J$ are induced by permutation representations of $(S_n)_{J_z}^z$. This would strengthen the preceding theorem, whi…

Science Fiction conjecture. The bigraded $\mathfrak{S}_n$-module $\bigcap_{i=1}^{k}V_{\mu^{(i)}}$ has dimension $\frac{n!}{k}$, and its Frobenius characteristic is

Butler's conjecture. The symmetric function $\operatorname{I}_{\lambda,\mu}[X;q,t]$ is Schur positive.

Let $\min\mathbb{N}$ and $lin\{0,1,\dotsc,m\}$. For partitions $u$ and $\tau$, let $\mathsf{A}{l,m}( u,\tau)$ be the sequence of plethysm coefficients defined from the associated skew…

Let $u$ and $\tau$ be arbitrary partitions, and let $minmathbb{N}$. The notation $\mathsf{A}{m-1,m}( u,\tau)$ denotes the associated sequence of plethysm coefficients. **Wildon's sta…

Let $\lambda$ be a partition, and let $P(s_\lambda\mapsto0)$ denote the probability that the Schur function $s_\lambda$ evaluates to zero over the finite field under consideration.…

Let $I_n$ be the ideal generated by alternating polynomials in $R=\mathbb{C}[x_1,y_1,\ldots,x_n,y_n]$, and let $q$ and $t$ record the two gradings. Write $s_1^n$ for the $n$-fold p…

Lapointe–Vinet's Jack-operator conjecture. The creation operators share the following properties:

Lapointe–Vinet's operator-identification conjecture. The creation operators satisfy