33 problems

Let $\Lambda$ range over superpartitions, and let $m_\Lambda$ denote the monomial basis of symmetric superfunctions. Equip the space of symmetric superfunctions with the scalar pro…

Let $\Lambda$ be a superpartition, let $D[\Lambda]$ be its diagram, and let $a_{\Lambda}(s)$ and $\ell_{\Lambda}(s)$ denote respectively the arm and leg statistics of a cell…

Let $J_\Lambda^{(1/\beta)}$ be a Jack superpolynomial, let $c_\Lambda^{\mathrm{min}}(\beta)$ be its minimal coefficient, and expand it in the monomial basis as … The integrality co…

Two-dimensional sigma models with $d4a9=(2,2)$ supersymmetry may involve standard chiral, twisted chiral, and semi-chiral multiplets. Sevrin–Troost conjecture. Most general sigma m…

Let $\mathfrak B_n$ be the group of signed permutations, let $SR^B_n$ be the type $B$ superspace coinvariant ring, and let $\mathcal{OP}^B_n$ be the family of signed ordered set pa…

Let ${\sf s}_\Lambda$ be the Schur-type functions in superspace obtained from Macdonald polynomials in superspace with parameters $q=t=1$, let ${\sf S}_\Lambda$ be the orthonormal…

Fields Institute conjecture. The doubly graded ${\mathfrak S}_n$-structure of $SR_n$ is given by

Fields Institute conjecture. The dimension of $SR_n$ is the ordered Bell number, and, as an ungraded ${\mathfrak S}_n$-module, $SR_n$ is the permutation representation on ordered s…

For $1\leq k\leq n$, let $\mathbb{V}_{n,k}$ be the smallest subspace of $\Omega_n[y_1,\dots,y_n]$ containing the superspace Vandermonde $\delta_{n,k}$, closed under…

For $1\leq k\leq n$, let $\delta_{n,k}\in\Omega_n$ be the superspace Vandermonde, and let $V_{n,k}$ be the smallest linear subspace of $\Omega_n$ containing $\delta_{n,k}$ and clos…

Let $d_j:\Omega_n\to\Omega_n$ be defined for $j\geq1$ by … and let $\delta_n\in\mathbb{C}[x_1,\dots,x_n]$ be the Vandermonde determinant. Let $SR_n$ be the superspace coinvariant a…

Let $\Omega_n$ be the superspace algebra with commuting variables $x_1,\ldots,x_n$ and anticommuting variables $\theta_1,\ldots,\theta_n$, acted on by the symmetric group…

Let $\mathbb{V}_n({\mathbf {a}})$ be the extension of the superspace module $V_n({\mathbf {a}})$ obtained by adjoining commuting variables $y_1,\dots,y_n$ and closing under polariz…

Unimodality conjecture. The matrix of coefficients of ${\mathrm {Hilb}}(R_n({\mathbf a});q,z)$ has unimodal rows and columns. This would extend the observed symmetry and unimodalit…

Let $n=k+r$ and let ${\mathbf a}=(k-1,k-1,\dots,k-1)$ be a length-$r$ sequence whose entries are all $k-1$. Let $\mathcal V_n({\mathbf a})$ be the smallest subspace containing the…

Macdonald superspace Pieri-rule conjecture. The Pieri rules are

Let $L^{(n|0)}$ be a closed superform on a supermanifold, with degree equal to the dimension of its bosonic submanifold, and let $\Omega(x,\theta)$ be the arbitrary function appear…

Let $k,n$ be integers with $k>1$ and $n>1$, and use the semicolon to separate the fermionic and bosonic parts of a superpartition. Almost rectangular bilinear identities conjecture…

Let $k,n$ be integers with $k>1$ and $n>1$. Let $r',r\in\{k,k-1,0\}$ satisfy $r'>r$, and set $\epsilon=\delta_{r,0}$. Here $k^n$ denotes the ordinary partition with $n$ parts equal…

Let $s_\Lambda$ be a Schur superpolynomial indexed by a superpartition, and let $(;r)$, $(r;)$, $(;1^r)$ and $(0;1^r)$ denote the corresponding bosonic and fermionic row and column…

A superpartition is denoted by $[?]$; let $s^c_\Lambda(x,\theta)$ be the combinatorially defined Schur superpolynomial and let $s_\Lambda(x,\theta)$ be the specialization at…

Let $F(x,\theta)$ be a polynomial in superspace of fermionic degree $m$, let $N\ge m$, and define … where $u_i=t^{i-1}/q^{\max(m-i,0)}$ and…

Let $P_\Lambda(x,\theta;q,t)$ be a Macdonald superpolynomial and let $h^\downarrow_\Lambda(q,t)$ be the product over bosonic boxes defined by … Define…

Let $\Lambda$ be a superpartition of fermionic degree $m$, and let $P_\Lambda(x,\theta;q,t)$ be the Macdonald superpolynomial. Let $\mathcal B\Lambda$ be the set of bosonic boxes,…

Generalized conjugation conjecture. Then