Kronecker delta return 1, or 0 depending on a conditional statement (if $i = j$), for example, $\delta_{i,j} = 1$ if $i = j$, and $\delta_{i,j} = 0$ otherwise. I would like to know if there are conventional symbols for similar expressions with conditional statements such as (if $i \lt j$) and (if $i\leq j$)?
For instance, a conventional symbol for $X$ in the following expression: $X_{i,j} = 0$ if $i \lt j$, and $X_{i,j} = 1$ otherwise.
Let's begin by talking about the Iverson bracket $[p]$, which for a claim $p$ is equal to $1$ if $p$ is true or $0$ if $p$ is false. Thus $\delta_{ij}=[i=j]$, while your $X_{ij}$ is $[i\ge j]$. I'm unaware of any other notation you could use (except, say, $\sum_{k\ge j}\delta_{ik}$ or $\sum_{k\le i}\delta_{jk}$).