In this short section we limit ourself to recall the definitions needed to state the Delta conjectures, in particular the definition of the Delta operators. For the missing notation we refer to \cite[Section~1]{DAdderio-VandenWyngaerd-2017} or \cite{Haglund-Book-2008}.

We denote by $\Lambda$ the algebra over the field $\mathbb{Q}(q,t)$ of symmetric functions in the variables $x_1,x_2,\dots$. We denote by $e_n$, $h_n$ and $p_n$ the \emph{elementary}, \emph{complete homogeneous} and \emph{power symmetric function} of degree $n$, respectively. We denote by $\omega$ the involution of $\Lambda$ defined by $\omega(e_n):=h_n$ for all $n$.

Also, for any partition $\mu$, we denote by $s_{\mu}\in \Lambda$ the corresponding \emph{Schur function}. It is well-known that the symmetric functions $\{s_{\mu}\}_{\mu}$ form a basis of $\Lambda$. The \emph{Hall scalar product} on $\Lambda$, denoted $\<\, ,\, \>$, can be defined by stating that the Schur functions are an orthonormal basis.

Let $\widetilde{H}_{\mu}\in \Lambda$ denote the \emph{(modified) Macdonald polynomial} indexed by the partition $\mu$. As the polynomials $\{\widetilde{H}_{\mu}\}_{\mu}$ form a basis of $\Lambda$, given a symmetric function $f\in \Lambda$, we can define the \emph{Delta operators} $\Delta_f$ and $\Delta_f'$ on $\Lambda$ by setting
%
\begin{align}
\Delta_f \widetilde{H}_{\mu} \coloneqq f[B_{\mu}(q,t)] \widetilde{H}_{\mu} \quad \text{ and } \quad 
\Delta_f' \widetilde{H}_{\mu}  \coloneqq f[B_{\mu}(q,t)-1] \widetilde{H}_{\mu}, \quad \text{ for all } \mu,
\end{align}
where $B_{\mu}(q,t)=\sum_{c\in \mu}q^{a_\mu'(c)}t^{l_\mu'(c)}$ ($a_\mu'(c)$ and $l_\mu'(c)$ are the coarm and coleg of $c$ in $\mu$, respectively) and the square brackets denote the \emph{plethystic} substitution: if $X = x_1 + x_2 + \dots$ is a sum of monomials, then $f[X] \coloneqq f(x_1, x_2, \dots)$. 

For example, if $\mu=(3,2)$ and $f=e_4$, then $B_{(3,2)}(q,t)=1+q+q^2+t+qt$, so that $f[B_\mu(q,t)]=e_4(1,q,q^2,t,qt)=q^4 t^2+q^4 t+q^3 t^2+q^3 t+q^2 t^2$ and $f[B_\mu(q,t)-1]=e_4(q,q^2,t,qt)=q^4t^2$.

%and both $B_\mu$ and $B_\mu - 1$ are, in fact, sum of monomials.

Finally, we recall here the notation for the \emph{$t$-analogue} of $n\in \mathbb{N}$:
\[[n]_t\coloneqq 1+t+t^2+\cdots +t^{n-1}. \]

 