23 problems

Let $J_\Lambda^{(1/\beta)}$ be a Jack superpolynomial, and write its supermonomial expansion in the form … Here $\Omega\leq\Lambda$ denotes the ordering used for the triangular exp…

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 $\Omega_n=\mathbb{C}[x_1,\ldots,x_n]\otimes\bigwedge\langle\theta_1,\ldots,\theta_n\rangle$ be the superspace, and let $SR^B_n$ be its type $B$ superspace coinvariant ring. Wri…

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…

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 $\Omega_n[y_1,\dots,y_n]=\mathbb{C}[x_1,\dots,x_n]\otimes\mathbb{C}[y_1,\dots,y_n]\otimes\bigwedge\{\theta_1,\dots,\theta_n\}$, with $S_n$ acting triply diagonally, and let … b…

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…

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 $k,n$ 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 $\Lambda$ 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

Let $\Lambda$ be a superpartition of fermionic degree $m=1$, let $H_\Lambda$ be the modified Macdonald superpolynomial, and let $\psi$ be the linear map defined on Schur superpolyn…

Concatenation conjecture. One has

Conjugation conjecture. The map $\psi$ satisfies

Let $P_\Lambda$ be the Macdonald superpolynomial associated with a superpartition $\Lambda$ of fermionic degree $m$, and let $h^\uparrow_\Lambda$ and $h^\downarrow_\Lambda$ be the…

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

Self-duality conjecture. For every $f\in {\bf{L}}_{{\mathfrak{B}}_{{L},\mathrm{ev}}}({\mathfrak{B}}_{{L},\mathrm{od}}:{\mathfrak{B}}_{{L}})$, there exists…