# Irreducible decomposition of tensor product of irreducible $S_n$ representations

Are there well known results on the irreducibles in the decomposition of tensor products of irreducible $S_n$ representations? I would also like to know of some references where I can find formulas (if they exist in the literature) for finding multiplicities.

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I recall having seen the multiplicities called "Kronecker numbers" or sth. like that, with the remark that they are way harder to compute than Littlewood-Richardson numbers. I do not have a free minute for searching for the reference, though. – darij grinberg May 13 '11 at 9:20
What you are looking for is the Kronecker product. It seems to be helpful to throw the word "Schur" into your search as well. These are much harder to compute than LR numbers, and not much is known. I don't know a good survey article; if someone does, I hope they'll post a link. – David Speyer May 13 '11 at 11:43
For a start, you could look into papers by Remmel and Whitehead. And for more recent work, you could try articles by Rosas, Orellana and Briand. Also, the thesis of Rosas could prove handy. Of course, they only tackle special cases. – Vasu vineet May 13 '11 at 13:21

## 1 Answer

The numbers you want are called Kronecker coefficients. Bürgisser and Ikenmeyer "The complexity of computing Kronecker coefficients" showed that they are hard to compute in general, so in particular there are no "easy" formulas for them. (There are some explicit formulas for simple special cases.)

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This is unconvincing. Compare with "On the complexity of computing Kostka numbers and Littlewood-Richardson coefficients" by Narayanan cs.uchicago.edu/~hari/kostka.pdf . This paper says that the complexity of computing LR-coefficients and even Kostka numbers is #P-complete. So what? We all know that Kronecker coefficients are much harder to deal with; in particular they have no (known) nice combinatorial interpretation in the general case. – Igor Pak May 14 '11 at 20:03
I also thought LR coefficients were easy to compute due to the proof of saturation conjecture by knutson and Tao. Am I wrong? I thought only kronecker numbers were hard. – Turbo Mar 21 '12 at 19:07