Which is the importance of Young’s tableaux in mathematics?

Question

I don’t know much about combinatorics, I’m just getting started on this. I want to know, why Young’s tableaux are important? and why it is important to relate them to matrices? Thank you very much.

Answer 1

The standard Young tableaux of partition $\lambda$ of $n$ correspond one to one with the irreducible representations of the symmetric group $S_n$ in the ordinary case (characteristic of field no divisor of group order).

Answer 2

Schur polynomials, which are an important family of symmetric functions, are defined using them. Jacobi-Trudi identity is a way to compute this polynomials using matrices.

Answer 3

Any polynomial representation of $GL(n,\mathbb{C})$ has a weight-space decomposition which is basically simultaneous eigenspaces for the diagonal matrices (i.e, maximal torus). Now a lot of information about the representation is encoded in the dimensions of these weight spaces. So we are interested in finding out the dimensions of these weight spaces. This is done by writing out the character of the representation. In particular since every representation can be decomposed as a sum of irreducible representations and the character then becomes the sum of characters, we want just to know the character of each irreducible representation. The finite dimensional irreducible polynomial representations of $GL(n,\mathbb{C})$ are in one to one correspondence with the partitions with atmost $n$ parts. Let us write $V(\lambda)$ for the irreducible representation corresponding to a partition $\lambda$. There exists several formulas for finding the character of $V(\lambda)$ for example the Weyl Character formula. But the problem with this is that it involves mixed signed sums, and can take quite a while to compute by hand. On the other hand, since all we are finding out is the dimensions of weight spaces, which are nonnegative integers we might ask ourselves is there a way to compute the dimensions combinatorially, i.e, in this case count something? And here comes the role of Young Tableux. The number of semistandard Young Tableux of shape $\lambda$ and weight $\mu$ is the dimension of the weight space $V(\lambda)_{\mu}$ and this also gives the combinatorial definition of Schur polynomial, which among other things are the characters of irreducible polynomial representations of $GL(n,\mathbb{C})$.

In addition, Young Tableux can be used to give the Littlewood-Richardson rule, which answers the question how does the tensor product of two irreducible representation of $GL(n,\mathbb{C})$ decompose as a sum of irreducibles.

And all this is just the tip of the iceberg.