Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
Have your read the first chapter of Kleshchev's "Linear and projective representations of symmetric groups"? This reminds me of ideas of Okounkov & Vershik front.math.ucdavis.edu/0503.5040. –  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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.