Young Tableau: A Monoid (Part III)

View literate file on Github

> import YT

Last time, we defined an operation that allowed us to grow a young tableau by a single number. It turns out that we can use this to give a monoid structure to the space of young tableaux called the Schensted operation.

The row-bumping lemma asserted that when inserting x_1 followed by x_2 into T (as in (T \leftarrow x_1) \leftarrow x_2) when x_1 \le x_2 the box introduced by x_1 is strictly to the left of and weakly below the box introduced by x_2. By extension, we arrive at the following proposition.

Proposition Suppose we constructed U = ((T \leftarrow x_1) \leftarrow x_2) \leftarrow \dots \leftarrow x_p where x_1 \le \dots \le x_p and U and T have shapes \mu and \lambda then no two boxes in \mu / \lambda are in the same column. Conversely, if no two boxes are in the same column in \mu / \lambda, then there is a unique tableau T of shape \lambda and unique x_1 \le \dots \le x_p such that U = ((T \leftarrow x_1) \leftarrow x_2) \leftarrow \dots \leftarrow x_p where x_1 \le \dots \le x_p.

Thus, we may define a product tableau T \cdot U from any two tableau T,U. And it turns out that this product has an identity (empty tableau) and is associative.

> productTableau :: Yt -> Yt -> Yt
> productTableau t =
>     foldl' (\y x -> fst $ rowInsertion x y) t . concat . reverse . yt
> instance Semigroup Yt where
>     (<>) = productTableau
> instance Monoid Yt where
>     mempty = Yt []
>     mappend = (<>)

Here is an example of a product and a QuickCheck to test that the operation is associative.

let yt1 = Yt [[1,2,2],[3]]
let yt2 = Yt [[3,5],[4]]
yt1  yt2


> prop_associative_schensted :: Property
> prop_associative_schensted = do
>   forAll (arbitrary::Gen Yt) $ \y1 ->
>     forAll arbitrary $ \y2 ->
>       forAll arbitrary $ \y3 -> y1 <> (y2 <> y3) == (y1 <> y2) <> y3

Chapter one of the book contains one more section that shows another construction (jeu de tarqin operation) to arrive at this monoid but I’ll skip it for now and move on to the next chapter where it looks at how this operation behaves on words (i.e. encodings of young tableaux).

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