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Let $n \in \mathbb{N}$ and let $\lambda \vdash n$, a partition of $n$. By a $\lambda$-tabloid I mean a row-tabloid of shape $\lambda$. There is a well-known order on the set of $\lambda$-tabloids, coming from an order on the set of compositions of $n$. It is a partial order and is also called the dominance order, see here for its definition on partitions.

Question: besides the dominance order, are there any other interesting partial orders on the set of tabloids?

The reason that I am interested in considering other partial orders, is because of the following lemma, which is Lemma 2.5.8. on page 69 of Sagan's book 'The Symmetric Group'. Here, $\mu$ is a partition of $n$ and $M^{\mu}$ is the $\mathbb{C}$-vector space spanned by all $\mu$-tabloids. We say that a tabloid $\{t\}$ appears in a vector $v = \sum_{T}c_{T}\{T\} \in M^{\mu}$, if the coefficient $c_{t} \neq 0$. In the lemma, the dominance order on the set of tabloids is used. However, it remains true for any partial order on tabloids.

Lemma: Let $v_{1},...,v_{m}$ be elements of $M^{\mu}$. Suppose, for each $v_{i}$ there is a tabloid $\{t_{i}\}$ appearing in $v_{i}$ such that

  1. $\{t_i\}$ is maximum in $v_i$, and
  2. the $\{t_i\}$ are all distinct.

Then $v_{1},...,v_{m}$ are linearly independent.

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  • $\begingroup$ I think the following linear order, which is much simpler, also good for the proof of the Lemma 2.5.8: The first integer in different rows of the two given tabloids appears first in the larger one. $\endgroup$ Dec 1, 2016 at 0:52

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