  Based on representations of the symmetric group $S_N$, explicit and exact
Schr\"odinger equation is derived for $U=\infty$ Hubbard model in any
dimensions with arbitrary number of holes, which clearly shows that during the
movement of holes the spin background of electrons plays an important role.
Starting from it, at T=0 we have analyzed the behaviour of the system depending
on the dimensionality and number of holes. Based on the presented formalism
thermodynamic quantities have also been expressed using a loop summation
technique in which the partition function is given in terms of characters of
$S_N$. In case of the studied finite systems, the loop summation have been
taken into account exactly up to the 14-th order in reciprocal temperature and
the results were corrected in higher order based on Monte Carlo simulations.
The obtained results suggest that the presented formalism increase the
efficiency of the Monte Carlo simulations as well, because the spin part
contribution of the background is automatically taken into account by the
characters of $S_N$.
