  Block character of a finite symmetric group S(n) is a positive definite
function which depends only on the number of cycles in permutation. We describe
the cone of block characters by identifying its extreme rays, and find
relations of the characters to descent representations and the coinvariant
algebra of S(n). The decomposition of extreme block characters into the sum of
characters of irreducible representations gives rise to certain limit shape
theorems for random Young diagrams. We also study counterparts of the block
characters for the infinite symmetric group S(\infty) along with their
connection to the Thoma characters of the infinite linear group GL(\infty,q)
over a Galois field.
