Let $n$ be a natural number. Consider a set $\Lambda_n$ of partitions of $n$ into a sum of natural numbers, like $n = \lambda_1 + \cdots +\lambda_k$ (A set of small lambdas representing a partition is considered to be unordered).

There is a natural partial order on $\Lambda_n$. Namely, we say that $\lambda \ge \lambda'$ if $\lambda$ is a refinement of $\lambda'$. For example for $n=3$ the partition $1+1+1$ is a refinement of the partition $2+1$, so $1+1+1 \ge 2+1$.

From the other hand we know, that elements of $\Lambda_n$ are in 1-to-1 correspondence with a set of elements of a certain basis of the centre of a complex group algebra of the symmetric group $S_n$. This centre has a nice structure of a commutative Frobenius algebra.

Is there a way to find out whether two partitions, represented as basis elements of the above mentioned algebra are comparable?

Edit: May be I was not very precise. I mean the following. Consider, we have a concrete commutative Frobenius algebra with a concrete basis. Now we want to describe some partial order on this basis in the inner terms of the algebra (product, trace) only. That problem arose in the following context. I wanted 1) to describe all possible partitions majorated by given 3 partitions, 2) to introduce an analogue of this partial order on an arbitrary finite group.

Why to use algebra formalism ? In fact I am working on the problem of counting certain combinatorial objects with weights, that can be computed as traces of monomials in the algebra. So, it is just a wild hope that this problems can be related.

  • $\begingroup$ Hi Max, I'm not sure I see which kind of answer you expect, since (1) it seems easy to see if two partition are comparable (2) the underlying partition $\mu$ of the basis element $C_{\mu}$ you mention is completely apparent, since $C_{\mu}$ is just the formal sum of all permutation of type $\mu$, so I don't understand in which sense this algebra should help ! $\endgroup$ – Adrien Mar 3 '12 at 18:14
  • $\begingroup$ Hi Adrien. I meant the following. Consider you know only the structural tensor and the trace on the algebra (you are working in a concrete basis; you have completely forgotten that the elements of this basis came from partitions). Is it possible to determine the partial order on the basis which coincides with the order induced from partitions in purely algebraic terms? In fact, I am trying to solve the next two questions: 1) find all possible partition which are majorated by given 3 partitions 2) Find an analogue of the described order for an arbitrary finite group. Algebra could help there. $\endgroup$ – Max Karev Mar 4 '12 at 22:04
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    $\begingroup$ Probably you already know that the "dominance" partial order is readily available through the Kostka numbers. $\endgroup$ – John Wiltshire-Gordon Mar 6 '12 at 7:58

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