Graded identities for the algebra of $n\times n$ upper triangular matrices over an infinite field

We consider the algebra $U_n(K)$ of $n\times n$ upper triangular matrices over an infinite field $K$ equipped with its usual $\mathbb{Z}_n$-grading. We describe a basis of the ideal of the graded polynomial identities for this algebra, and compute some of the numerical invariants of this ideal. An extended version of this research will be published elsewhere.