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Is there a standard notation that expresses substructure? The specific case that I care about is the following:

Suppose $\sigma,\tau$ are permutations such that $$\sigma(x)\not=x\implies \sigma(x)=\tau(x) \qquad (1)$$ This is equivalent to saying that the cycles appearing in the cycle decomposition of $\sigma$ are a subset of the cycles appearing in the cycle decomposition of $\tau$. Since I am not aware of a special notation to describe this condition, and I use this type of condition frequently in a paper I am working on, I have been using $\sigma\subset\tau$ as a shorthand.

Another way I have been thinking of describing this property is via a projection. For instance, $\sigma\subset\tau$ is the same as saying $\sigma=\tau|_{S'}$ where $S'$ is a suitable embedding of a smaller symmetric group.

When condition $(1)$ is used, it seems like the proper generalization is some sort of "refinement" or "continuation" condition, like saying $\text{supp}(f)\subset\text{supp}(g)$ and $f=g$ on the common support.

So there are three ways of thinking about this property:

  1. Information about a decomposition
  2. Restriction to a subspace
  3. Continuation of a function

Is there a common notation that is used in these three cases? Are there conditions for other kinds of objects that also have interpretations along all these lines?

Just looking for some insight.

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Have your read the first chapter of Kleshchev's "Linear and projective representations of symmetric groups"? This reminds me of ideas of Okounkov & Vershik – David Hill May 1 '13 at 1:10
Very interesting; this is all news to me. Thanks for pointing me in this direction! – pre-kidney May 1 '13 at 1:42
In the spirit of my school, I would note $Cycl(\sigma)$ for the set of cycles of $\sigma$. – Duchamp Gérard H. E. May 1 '13 at 3:40
Can you post a link to a reference where this notation is used? – pre-kidney May 6 '13 at 6:54

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