Subordinate and Superordinate Frames
Point of Post: In this post we define the notion of a subordinate frame and superordinate frame and discuss equivalent ways of defining them.
It’s clear that in our definition of -frames that sitting inside each -frame is a lot of -frames which can be gotten simply by removing a single box from . These -frames ‘sitting’ inside shall be what we call the -frames ‘subordinate’ to . Of course, there is a dual notion where given an -frame we see that sits subordinately inside a lot of -frames , we shall say in this case that is ‘superordinate’ to . Said slightly differently the -frames superordinate to are the -frames which can be obtained from by adding a single box to . The interesting thing is that given (the number of standard Young tableaux) for each subordinate to we can calculate and dually given for all -frames superordinate to we can calculate . That will be the topic of our next post
Let be an -frame, then we say that the -frame is subordinate to , written , if can be obtained from by removing one box. For example,
Given a frame we denote, when convenient, the set of all frames subordinate to by . If given a specified block we denote the collection of squares obtained by removing from by . It’s clear then that
A square in a -frame is called a bottom right corner of if the square has no square to the right or below it. If is given in the row structure notation as then a square is a bottom right corner if it occurs as the square in the row (for some ) and either or . We denote the set of all bottom right corners of by . What we note is that the only removable squares occur as bottom right corners in the sense that if a square is removable then there must be no squares to the right or below that square otherwise the resulting collection of squares (after removing the square in question) shall not be a Ferrer’s diagram. Moreover, it’s clear that if a square occurs as a bottom right corner in then removing that square results in a Ferrer’s diagram and so we arrive at the following theorem:
Theorem: Let be an -frame. Then,
In particular, .
We now define the dual notion of a subordinate frame. Namely, if is a -frame and is a -frame which can be obtained by adding one block to we say that is superordinate to , and denote this . We denote, when convenient, the set of all -frames superordinate to a -frame by . It’s evident that if is a -frame that
Let be a -frame with row structure . It’s evident that we may insert a square at the end of the row if and only if or . It thus makes sense that
where the last term represents the -frame obtained from by forming a new row. Note though that each which is not the -frame obtained by starting a new row corresponds uniquely to a block at the end of a row of with or . But, each of these blocks is either an element of or lies directly above an element of . For example, the block marked in the following -frame
is a block for which one may add a block beside it to obtain a -frame superordinate to and yet this block is not an element of we see that it corresponds (uniquely) to the bottom right corner marked below
it thus follows that
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.