Consider the problem of n cards labelled 1 through n, lying face up on a table. Suppose two integers a and b are chosen independently and uniformly between 1 and n. The cards labelled a and b are switched. If many such transpositions are made the row of cards will tend to appear in random arrangement. Then question is how many steps are required until the deck is well mixed up (i.e. the permutation is close to random)? Diaconis and Shahshahani used tools of representation theory of symmetric groups to prove that at least 1/2(n log n) steps are required before the deck will be well mixed up for large n. I will explain ideas of their proof and few related problems.

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