45 problems

For $k>0$, let $A_\lambda^{(k)}[X;t]$ denote the proposed $k$-atom and let $s_\lambda^{(k)}$ denote the $k$-Schur function. Lapointe–Morse conjecture. The family…

Non-commutative $k$-Schur positivity conjecture. The coefficients $b_{w,v}$ are positive, namely

Let $H_\mu(x;q,t)$ be the modified Macdonald polynomial and write its expansion in the $k$-Schur basis as … with…

k-Schur multiplication positivity conjecture. The structure constants satisfy

Let $\operatorname{Gr}$ be the affine Grassmannian, and let affine Schur functions $\tilde{F}_\lambda$ be indexed by the corresponding Grassmannian affine permutations. Morse–Shimo…

Intermediate-basis positivity conjecture. The coefficients satisfy

k-Schur Pieri conjecture. One has

Let $\lambda$ be a $k$-bounded partition, and let $s_\lambda^{(k)}[X;t]$ be the corresponding $k$-Schur function. The coproduct positivity conjecture. There are polynomials…

Let $\lambda$ be a $k$-bounded partition, let $\lambda^{\omega_k}$ be its $k$-conjugate, and define $\omega_t f=(\omega f)|_{t\to1/t}$. The $t$-involution conjecture. For every $k$…

Let $\lambda$ be a $k$-bounded partition and let $\ell$ be a positive integer with $\ell\leq k$. Define $E_{\lambda,\ell}^{(k)}$ and $\bar E_{\lambda,\ell}^{(k)}$ as the index sets…

For $k$-bounded partitions $\lambda$, $\mu$, and $\nu$, write the product of $k$-Schur functions as … where $c_{\lambda\mu}^{\nu}$ are the ordinary Littlewood–Richardson coefficien…

Let $\lambda$ be a $k$-bounded partition, meaning that its largest part is at most $k$. Let $\lambda^{\omega_k}$ denote its $k$-conjugate, and let $\omega$ be the usual involution…

Let $s_\lambda^{(k)}[X;t]$ denote the $k$-Schur functions and let the connection coefficients in their expansion against the relevant Macdonald polynomials be integral polynomials…

For each $k$-bounded partition $mu i ttext{Par}^k$, let $tilde{A}^{(k)}mu$ be the introduced family of symmetric functions. $k$-split $k$-Schur conjecture. The family…

Let $(Psi,mu)$ be an indexed root ideal, with $mu i ttext{Par}^k$, and define $operatorname{band}(Psi,mu)i$ as in the paper. $k$-Schur positivity conjecture. If…

The $k$-split polynomials and strong tableaux provide two definitions of $k$-Schur functions. Agreement conjecture. The $k$-Schur functions defined via $k$-split polynomials agree…

Haiman–Hugland's conjecture. The symmetric function

Let $\lambda$ be a partition, let $\operatorname{kskew}(\lambda)$ denote its associated $k$-skew diagram, and let $\Phi(\operatorname{kskew}(\lambda))$ be the Chen-Haiman root idea…

Let $k$-Schur functions be indexed by partitions $u$ with $u1leq k$. Chen-Haiman constructed an ideal of positive roots associated to each such partition, and let…

Strong-order interval conjecture. For every $la\in P_k$ and $1\le t\le k$, there exists $nu\in P_k$ such that

Interval conjecture. For every $4\lambda\in\mathcal{P}_k4$ and $1\le t\le k$, there exists $4\mu\in\mathcal{P}_k4$ such that

Nonnegativity conjecture. For every $4\lambda\in\mathcal{P}_k4$ and every $\mu$, one has $a_{P,\lambda,\mu}\ge 0$.

Morse's conjecture. If $\lambda\subset R_i$ for some $1\leq i\leq k$, then

Let $A$ be an affine Bruhat counter-tableau, and let the statistic on $k$-tableaux discovered by Lapointe and Pinto be the statistic referred to in the source. Lapointe–Pinto stati…

Let $\lambda$ be a $(k+1)$-core with $deg(\lambda)=n$, and let $A$ range over affine Bruhat counter-tableaux of weight $1^n$ and inner shape $\lambda$. Let $spin^k(A)$ be the stati…