# Lattice structure (wrt dominance order) on the set of Young diagrams appearing in the decompositions given by the Littlewood-Richardson rule

The irreducible decomposition of the tensor product of two irreducible representations of GL(n) is described by the Littlewood-Richardson rule. This same rule also governs the decomposition of the product of two Schur polynomials into a linear combination of Schur polynomials. In both cases, we label the components of the product and the decomposition with Young diagrams, or integer partitions.

We noticed that the Young diagrams in the decomposition always seem to form a lattice w.r.t. dominance order. That is, for each pair of diagrams in the decomposition, the least upper bound ("join") and the greatest lower bound ("meet") of the pair is also part of the decomposition.

For example, looking at the following decomposition (where the Young diagrams are denoted by the corresponding integer partitions): $$(2,1) \otimes (2,1)=(4,2) \oplus (4,1,1) \oplus (3,3) \oplus 2 (3,2,1) \oplus (3,1,1,1) \oplus (2,2,2) \oplus (2,2,1,1)$$ The two unordered pairs in this decomposition are: (2,2,2), (3,1,1,1); and (3,3), (4,1,1). The join and meet of the first pair are (3,1,1,1) and (2,2,1,1) respectively. For the second pair, these are (4,2) and (3,2,1). In both cases, the join and meet diagrams are parts of the decomposition. We checked it for many larger examples with extensive symbolic calculations, and the product had this lattice property in all cases.

Thus my question is: Applying the Littlewood-Richardson rule to a pair of Young diagrams, does the set of Young diagrams appearing in the result (with nonzero multiplicity) always form a lattice wrt dominance order?

• Possibly this follows from one of the polytopal models of the LR coefficients... Sep 11, 2021 at 20:47
• Thanks, we checked some of these models, and it has some relations to some other important questions in the project of ours. Sep 13, 2021 at 20:49

• Thank you very much, this helped! The cited paper in the link (O. Azenhas, "The admissible interval for the invariant factors of a product of matrices") proves the following: Considering two fixed Young diagrams $\mu$ and $\lambda$, the Young diagrams $\nu$ with Littlewood-Richardson coefficients satisfying $c_{\mu, \nu}^{\lambda}>0$ form a sublattice wrt dominance order. This is very much related but not exactly what we asked for in the question… continued in the next comment Sep 13, 2021 at 20:45
• Continued… Our conjecture was that if we fix two Young diagrams $\mu$ and $\nu$, then the Young diagrams $\lambda$ with Littlewood-Richardson coefficients satisfying $c_{\mu, \nu}^{\lambda}>0$ form a sublattice. Nevertheless, using some symmetries of the Littlewood-Richardson coefficients (derived from representation theory, not combinatorics), we managed to show that the two statements are equivalent. Sep 13, 2021 at 20:45
• See also the paper by Hugh Thomas and Alex Yong concerning symmetry. Sep 14, 2021 at 5:34

Olga Azenhas' paper does not prove that for two fixed Young diagrams $$\mu$$ and $$\lambda$$, the Young diagrams $$\nu$$ with Littlewood-Richardson coefficients satisfying $$c_{\mu,\nu}^\lambda>0$$ form a sublattice w.r.t. dominance order. On p95-96 she even gives an example of where this fails: $$s[5^2,4^2,1]/[3,2]= s[4^2, 3, 2, 1] + s[4^2, 3, 3] + s[4^3, 2] + s[5, 4, 2^2, 1] + s[5, 4, 3, 1^2] + 2s[5, 4, 3, 2] + s[5, 4^2, 1] + s[5^2, 2, 1^2] + s[5^2, 2^2] + s[5^2, 3, 1].$$ Notice that the Schur functions $$s[4^3,2]$$ and $$s[5,4,3,1^2]$$ each appear with multiplicity $$1$$. Now the meet of $$[4^3,2]$$ and $$[5,4,3,1^2]$$ is $$[4^3,1^2]$$, but $$s[4^3,1^2]$$ appear with multiplicity $$0$$ in the expansion of $$s[5^2,4^2,1]/[3,2]$$. Even though this does not directly answer your original question, I think it indicates that the answer is very likely to be no: products of Schur functions do not have the lattice property.

• So the theorem is that we have a subposet of dominance order, and this subposet is a lattice, but it is not a sublattice of dominance order! Thank you for clarifying! Jun 8 at 14:35