84 problems

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.

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

Lapointe–Vinet's creation-operator conjecture. The Macdonald polynomials are given by

Let $n$ be a positive integer, let $g$ and $ell$ be parameters, and let $\lambda=(\lambda_1,\ldots,\lambda_n)$. Write $(y;q)_m$ for the $q$-Pochhammer symbol and…

Three-variable basic hypergeometric conjecture. The series

Let $P_{\lambda}(x;q,t)$ denote the Macdonald polynomial corresponding to a partition $\lambda$, and let … where the sum runs over all partitions $\lambda$. Write $p_k$ for the pow…

Let $n\geq 0$, let $\mu\overline{\nu}$ be a dominant weight of $U(2n)$, and let $Z$ normalize the integral so that the trivial polynomial integrates to $1$. Nonstandard $U(2n)$ van…

Let $n\geq 0$, let $\mu\overline{\nu}$ be a dominant weight of $U(2n)$, and let $Z$ normalize the integral so that the trivial polynomial integrates to $1$. $U(2n)$ Grassmannian co…

Let $m$ and $n$ be integers with $0\leq m\leq n$, let $\mu\overline{\nu}$ be a dominant weight of $U(m+n)$, and let $P_{\mu\overline{\nu}}$ be the corresponding Macdonald polynomia…

Let $n$ be a positive integer, and let $mu$ and $\lambda$ be partitions of $n$. Let $m_\mu$ be the monomial symmetric function, $s_\lambda$ the Schur function, and $\nabla$ the Berg…

Let $p_\mu$ be a power-sum symmetric function, let $P^\circ_\nu$ be the shifted power-sum basis, and define … where $\operatorname{sh}_{q,t}$ is Lassalle's Macdonald shift map. Pow…

Let $(\mathbf {U},\mathbf{B},\gamma)$ be a commutative triple, and suppose that the action of $W_\Sigma$ on $\mathfrak{B}(\gamma)$ is transitive and that one stabiliser $W_J$ is a…

Let $\lambda$ and $\mu$ be partitions, and let $P_\lambda(x;q,t)$ denote the Macdonald polynomial. Macdonald containment conjecture. The Macdonald counterpart over…

Let $\lambda$ and $\mu$ be partitions with $|\lambda|=|\mu|$. Let $P_\lambda(x;q,t)$ be the Macdonald polynomial, $\mathcal M_S(\mathbb F_{\geqslant0})$ the Muirhead semiring over…

Monical–Tokcan–Yong's conjecture. If $\beta\in\widehat{\mathcal{P}}_{\alpha}$ and $\beta\in\mathbb{Z}^n_{\geq 0}$, then $\beta\leq_{S}\alpha$.

Monomial-product conjecture.

Rectangular multi-$t$ Yang–Baxter trace conjecture. The equivariant trace of $Y_{c3_bc}(v_bc)$ is proportional to

Let $n$ be a positive integer, let $\nabla_\ast$ be the operator on symmetric functions in two alphabets defined by…

Let $\beta$ be a weak composition, let $\mathsf{RTT}_0(\beta)=\bigcup_{\alpha\vDash_0\lvert\beta\rvert}\mathsf{RTT}_0(\alpha,\beta)$, and let $\operatorname{oldXi}$ be the symmetri…

The positivity conjecture. Let $\pi,\mu,\nu$ be three partitions. Then

The Macdonald generalization of the Matchings-Jack conjecture. For any positive integer $n$ and partitions $\pi,\mu,\nu$ of $n$,

Macdonald version of Stanley's conjecture. For any partitions $\lambda,\mu,\nu$, the quantity

The positivity conjecture. The quantity

Let $\lambda$ and $\mu$ be partitions with $\lambda\supseteq\mu$, and let $B_{\lambda\mu}$ be the integral binomial coefficient for the interpolation-polynomial family $\mathcal F$…