54 problems

A partial complex Hadamard matrix $H\in M_{M\times N}(\mathbb T)$ is regular when the scalar products between its rows decompose as sums of cycles. A Butson matrix is a complex Had…

Let $K$ be a nice knot, let $F_K(x,q)$ be its $F_K$ series, let $N_p(K,x)$ be its ADO polynomial, let $\Delta_K(x)$ be its Alexander polynomial, and let $\zeta_p=e^{2\pi i/p}$ be a…

Cyclic-sum conjecture. For every integer $n>r$ and every primitive root of unity $\zeta_n$, the following hold. With $r=\sum_{j=0}^{t}d_j+2t$,

For a quantum group $U_q(sl_2)$ at a root of unity, consider its semisimplicity and positivity properties. Semisimplicity–positivity conjecture. Semisimplicity and positivity shoul…

Conjecture 3. The elements $C_q$, $T_1^n$, $T_2^n$ and $C^{(n)}(I)$ generate the center of $U'_q({\rm iso}_2)$.

Conjecture 2. The central elements of [9] and of Theorem 4 generate the center of $U'_q({\rm so}_m)$.

Conjecture 1. The elements $C_q$ and $C^{(n)}(I_j)$, $j=1,2,3$, generate the center of $U'_q({\rm so}_3)$.

Let $q=\exp(\pi i/\ell)$ with $\ell\ge3$, and let $\operatorname{Conf}_{\vec\Lambda}^{(\ell)}$ be the space spanned by the lattice restrictions of the Weyl anti-symmetrized hyperge…

Let $Z$ be an infinite subset of the set of all roots of unity over $\mathbb{Q}$, and let … be the homomorphism induced by evaluating polynomials at the elements of $Z$. Injectivit…

Let $N>1$, and consider multiple polylogarithms evaluated at $N$-th roots of unity. The double-shuffle and distribution relations are families of relations among these values. Prim…

Let $N\geq 3$ be the order of a primitive root of unity, let $z_j^S(\mu)$ denote the centres of complete strings, and let $z_j^S$ be their values as $\mu\to1$. Let $n_S$ be the num…

Universal BRST-element conjecture. The element $Q$ can be used for any group to understand the structure of the tensor product at roots of unity and to define a new associative ten…

Centrality conjecture. The generators corresponding to real root vectors raised to the $\ell$-th power should be central.

Let $d\in\mathbb{N}_{\geq 2}$, let $\zeta$ be a primitive $d$th root of unity, and let $A_d$ be the set defined earlier in the paper. Let $\mathcal H(n,2\zeta)$ denote the correspo…

Safronov's root-of-unity skein conjecture. There is a line bundle $\mathcal{L}_q$ on $\mathrm{Loc}_{G^\vee}(M)$ such that

Let $\omega$ be a primitive $p$-th root of unity with $p\geq 1$, let $K$ be a nice knot, and let $F_K(x,q)$ be its $F_K$ series. Let $P_{K,\omega}^{(j)}(x)$ be the polynomials occu…

Let $K$ be a knot, let $J_K$ be its colored Jones function, let $\omega$ be a primitive $p$-th root of unity with $p\geq 1$, and let $\Lambda$ be the source's ring containing $J_K$…

Let $X$ and $X^!$ be symplectically dual varieties arising from Coulomb-branch and resolved Higgs-branch constructions for the gauge theory $(G,N)$. Let $\zeta$ be the relevant roo…

Let $p$ be a prime, let $n\geq 2$, and let $\zeta_{p^n}$ be a primitive $p^n$-th root of unity. For an element $\alpha\in \bbL_p$, write $[C_{\frac{1}{p-1}}(\alpha)]$ for the coeff…

Let $k$ be even, let $\zeta$ be a primitive $2k$-th root of unity, and let $Q_k$ denote the set of quasipolarities. For each permutation $\pi\in Q_k$, consider the product over…

Let $n>1$ be an odd integer, let $S_n$ be the symmetric group on $\\{1,\ldots,n\\}$, and let $D(n-1)$ be the set of permutations $\tau$ of $\\{1,\ldots,n-1\\}$ with $\tau(j)\ne j$…

Let $m$ and $n$ be positive integers, let $q$ be a nonzero complex number, and let $Q_{m,q}(n)$ and $Q_m(n,\mathbb{C})$ denote the $q$-deformed and ordinary $m$-quasi-invariant pol…

Pilehrood–Pilehrood–Tauraso's conjecture. For any positive integer $n$ with $n>k$, one has

Let $n>1$ be an integer, let $\zeta$ be a primitive $n$-th root of unity, and define the derangement set … For a permutation $\tau$, let $\operatorname{sign}(\tau)$ denote its sign…

Assume $n=3$, $\gcd(a_1,a_2,a_3)=1$, and let $\mathcal{A}_p=\{a_1\bmod p,a_2\bmod p,a_3\bmod p\}$ for $p=2,3$. Let $\zeta$ be a primitive cube root of unity, and let…