  A family of vectors $A \subset [k]^n$ is said to be intersecting if any two
elements of $A$ agree on at least one coordinate. We prove, for fixed $k \ge
3$, that the size of a symmetric intersecting subfamily of $[k]^n$ is $o(k^n)$,
which is in stark contrast to the case of the Boolean hypercube (where $k =2$).
Our main contribution addresses limitations of existing technology: while there
is now some spectral machinery, developed by Ellis and the third author, to
tackle extremal problems in set theory involving symmetry, this machinery
relies crucially on the interplay between up-sets and biased product measures
on the Boolean hypercube, features that are notably absent in the problem at
hand; here, we describe a method for circumventing these barriers.
