The uniqueness of plethystic factorisation
Chris Bowman, Rowena Paget

TL;DR
This paper proves the unique factorization property of plethysm products of Schur functions and classifies certain types of these products, advancing understanding in algebraic combinatorics.
Contribution
It establishes the first proof of unique factorization for plethysm products and classifies homogeneous and indecomposable cases.
Findings
Plethysm products of Schur functions can be uniquely factorized.
Classification of homogeneous and indecomposable plethysm products.
Provides foundational results for algebraic combinatorics and representation theory.
Abstract
We prove that a plethysm product of two Schur functions can be factorised uniquely and classify homogeneous and indecomposable plethysm products.
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The uniqueness of plethystic factorisation
Chris Bowman
and
Rowena Paget
Introduction
Let and denote the Schur functions labelled by the partitions and . There are three ways of “multiplying” this pair of functions together in order to obtain a new symmetric function; these are the Littlewood–Richardson, Kronecker, and plethysm products. The primary purpose of this paper is to address the most fundamental question one can ask of such a product: “does it factorise uniquely?”. For the Littlewood–Richardson product, this question was answered by Rajan [Raj04]. We solve this question for the most difficult and mysterious of these products, the plethysm product (which we denote ) as follows.
Theorem A**.**
Let be arbitrary partitions. If then either and ; or we are in one of five exceptional cases,
[TABLE]
In general, the decomposition of a plethysm product will have very, very many constituents. We ask: “when is the plethysm product of two Schur functions indecomposable?”. We prove that in fact such a product is always decomposable, and even inhomogeneous, except for some obvious exceptions. The analogous result for the Kronecker product was obtained by Bessenrodt and Kleshchev [BK99].
Theorem B**.**
Let be partitions. The product is decomposable and inhomogeneous except in the following exceptional cases:
[TABLE]
Understanding and decomposing the Kronecker and plethystic products of pairs of Schur functions was identified by Richard Stanley as two of the most important open problems in algebraic combinatorics [Sta00, Problems 9 & 10]. Almost nothing is known about general constituents of plethysm products; however the maximal terms in the dominance ordering are now well-understood [PW]. Our proof of Theorems A and B proceeds by careful analysis of these maximal terms.
Outside of combinatorics, plethysm products arise naturally in the representation theory of symmetric and general linear groups. In quantum information theory, the positivity of constituents in a plethysm product of two Schur functions is equivalent to the existence of quantum states with certain spectra, margins, and occupation numbers [AK08, BCI11]. Decomposing Kronecker and plethystic products of Schur functions is the central plank of Geometric Complexity Theory, an approach that seeks to settle the P versus NP problem [MS01]; this approach was recently shown to require not only knowledge of the positivity but also precise information on the actual multiplicities of the constituents of the products [BIP19].
1. Partitions, symmetric functions
and maximal terms in plethysm
We define a composition to be a finite sequence of non-negative integers whose sum, , equals . If the sequence is weakly decreasing, we say that is a partition and write . Given a partition of , the Young diagram is defined to be the configuration of nodes
[TABLE]
We say that a partition is linear if it consists only of one row, or one column. The conjugate partition, , is the partition obtained by interchanging the rows and columns of . The number of non-zero parts of a partition, , is called its length, ; the size of the largest part is called the width, ; the sum of all the parts of is called its size.
Given two partitions and , we let and denote the partitions obtained by adding the partition horizontally and vertically respectively. In more detail
[TABLE]
and is the partition whose multiset of parts is the disjoint union of the multisets of parts of and . We have that
[TABLE]
Finally we remark that, in this paper, the partition is usually equal to
[TABLE]
In other words, we often do not need to reorder the multisets of parts — this is simply because in most cases.
We now recall the dominance ordering on partitions. Let be partitions. We write if
[TABLE]
If and we write . The dominance ordering is a partial ordering on the set of partitions of a given size. This partial order can be refined into a total ordering as follows: we write if
[TABLE]
We refer to as the lexicographic ordering. We now define the transpose-lexicographic ordering as follows:
[TABLE]
We emphasise that this total ordering is not simply the opposite ordering to the lexicographic ordering; minimality with respect to is not equivalent to maximality with respect to .
Let be a partition of . A Young tableau of shape may be defined as a map Recall that the tableau is semistandard if and for all . We let for . We refer to the composition as the weight of the tableau . We denote the set of all tableaux of shape by , and the subset of those having weight by . The Schur function , for a partition of , may be defined as follows:
[TABLE]
The plethysm product of two symmetric functions is defined in [Sta99, Chapter 7, A2.6] or [Mac15, Chapter I.8]. The plethysm product of two Schur functions is again a symmetric function and so can be rewritten as a linear combination of Schur functions:
[TABLE]
such that . We say that the product is homogeneous if there is precisely one partition, , such that ; we say that the product is indecomposable if, in addition, . We now recall the role conjugation – often called the involution – plays in plethysm (see, for example, [Mac15, Ex. 1, Chapter I.8]). For , , and we have that
[TABLE]
Throughout this paper we shall let be partitions of and respectively. In order to keep track of the effect of this conjugation when comparing products and , we set
[TABLE]
Given a total ordering, , on partitions we let
[TABLE]
denote the unique partition, , such that and for all . We shall use this with both the lexicographic and transpose-lexicographic orderings . By equation 1.1 we have that
[TABLE]
The following theorems will be incredibly important in our arguments.
Theorem 1.1** ([PW, Corollary 9.1] and [Iij]).**
Let , be partitions of and respectively. The unique maximal terms of in the lexicographic and transpose lexicographic ordering are as follows :
[TABLE]
[TABLE]
Moreover, we have that
[TABLE]
Example 1.2**.**
When , Theorem 1.3 shows that
[TABLE]
Sometimes we shall use the dominance ordering to compare the summands of , and then there will, in general, be many (incomparable) maximal partitions. To understand these summands, we require some further definitions. We place a lexicographic ordering, , on the set of semistandard Young tableaux as follows. Let be semistandard -tableaux, and consider the leftmost column in which and differ. We write if the greatest entry not appearing in both columns lies in . Following [dBPW, Definition 1.4], we define a plethystic tableau of shape and weight to be a map
[TABLE]
such that the total number of occurrences of in the tableau entries of is for each . We say that such a tableau is semistandard if and for all . We denote the set of all plethystic tableaux of shape and weight by by .
Theorem 1.3** ([dBPW, Theorem 1.5]).**
The maximal partitions in the dominance order such that is a constituent of are precisely the maximal weights of the plethystic semistandard tableaux of shape . Moreover if is such a maximal partition then is equal to .
Finally, we recall the one known case in which every term in a plethystic product is both maximal and minimal in the dominance ordering. Given a partition of with distinct parts, we let denote the unique partition of whose leading diagonal hook-lengths are and whose row has length for . (An example follows.) We have the decomposition
[TABLE]
where the sum is over all partitions of into distinct parts. This decomposition is given in [PW16, Corollary 8.6] and [Mac15, I. 8, Exercise 6(d)]. We observe that for this product is never homogeneous (for example and both label summands).
Example 1.4**.**
For the decomposition obtained is
[TABLE]
We picture these partitions (and the manner in which they are formed) in Figure 3 below. We remark that
[TABLE]
by equation 1.1 simply because is even.
2. Decomposability and homogeneity of plethysm
In this section, we prove Theorem B of the introduction: namely we classify all decomposable/homogeneous plethystic products of Schur functions. This also serves to remove the homogeneous products from consideration in the proof of Theorem A.
Theorem 2.1**.**
Let be partitions of and , respectively. The product is decomposable and inhomogeneous except in the following cases:
[TABLE]
Proof.
That the listed products are homogeneous is obvious. We assume that and
[TABLE]
We shall show that this implies that and . We first assume that is non-linear, that is is neither nor . We set . We draw a horizontal line across the Young diagrams of and so that the partitions below each of these lines each have strictly fewer than nodes in total and are maximal with respect to this property. For , this line is drawn between the and rows (even though the row might be zero). For , this line is drawn at some point after the row. Since for , we see that as required.
It remains to consider the case that is linear and we assume (by conjugating if necessary) that . Then, as we saw in Example 1.2,
[TABLE]
Therefore row of has length which is at most 1, and row of has length at least . Since we are considering only , we conclude that and , that is . From the closed formula for the decomposition of in equation 1.2, and the resulting decomposition of its plethystic conjugate , we observe that the product is homogeneous if and only if . ∎
Corollary 2.2**.**
If or then either: and is a partition of 2; or at least one of or has size 1.
Therefore in the remainder of the paper, we can and will assume that none of the indexing partitions in our plethystic products are equal to .
3. Unique factorisation of plethysm
A quick scan of the diagrams in Figure 1 tells us that the maximal terms in the product under the lexicographic and transpose-lexicographic orderings encode a great deal of information concerning the multiplicands of the product. We might even think that these maximal terms are enough to uniquely determine the multiplicands. In fact, this is not the case (as the following example shows).
Example 3.1**.**
Consider the plethysm products
[TABLE]
Both have the same maximal terms in the lexicographic and transpose-lexicographic orderings, namely those labelled by and . Figures 4 and 5 depict how these two partitions can be seen to be maximal in the lexicographic and transpose-lexicographic orderings using Theorem 1.1.
This puts a scupper on our plans to determine uniqueness solely using maximal terms in the lexicographic and transpose-lexicographic orderings. Now, we notice that the plethysm products and can still be distinguished by looking at the maximal terms for both products in the dominance ordering. For example, labels a maximal term that appears in but it is not a maximal term in and . Similarly, labels a maximal term in but not in .
Our method of proof will proceed to distinguish plethysm products by first using maximal terms in the lexicographic ordering and only when necessary considering the broader family of terms which are maximal in the dominance ordering. We first consider the case where consists of a single row.
Theorem 3.2**.**
Let be partitions of respectively. We suppose that . If
[TABLE]
then either and or we are in the exceptional case
[TABLE]
Proof.
From the set-up, we know . We set for some . By assumption, we have that
[TABLE]
As a warm-up, we first consider the case where is linear. If and then (see Example 1.2) equation 3.2 says that . By comparing widths we deduce that . This implies and then . Now, suppose that and . Then which, as and has size , has final column of length 1. For equation 3.1 to hold, the same to be true of ; this implies . Similarly, comparing the final columns of and also shows that . Hence and we obtain a contradiction from comparing the widths of and .
We now assume that is non-linear so and . By equation 3.2,
[TABLE]
Since and , it follows that and, as , .
If then the left hand side of equation 3.3 is . Since , comparing the width in equation 3.3 shows that and that . This implies that is a hook partition whereas has second row of width at least , a contradiction.
Therefore we can assume that . Then equation 3.3 implies that the first rows of are all equal to and therefore for some . In particular, . We now consider equation 3.1: the difference between the first and second rows of is
[TABLE]
whereas the difference between the first and second rows of is less than or equal to . Therefore the necessary inequality
[TABLE]
implies that (since ). For the remainder of the proof and and therefore and .
We first consider the case . Here we have that and so the difference between the first and second rows of is . On the other hand, for the difference is at least . For equality, we require , that is . Then equation 3.1 becomes and we find . We now employ the dominance ordering to examine the case
[TABLE]
A necessary condition for is that . To see this, simply note that if , then
[TABLE]
and the maximum number of entries equal to 1 or 2 in a semistandard Young tableaux of shape is equal to (the sum of the lengths of the first and second rows of ). Thus for any such that by Theorem 1.3. We shall now construct a plethystic tableau with . This tableau will either be of maximal possible weight or there exists another plethystic tableau of the same shape but of weight ; in either case, for a partition for , whereas (by Theorem 1.3), providing us with the necessary contradiction. Let be the plethystic tableau such that
[TABLE]
This tableau has weight with and and so as required.
Finally, we consider the case . Here and is either or . In the case, comparing the widths of the partition on the left and right of equation 3.1 we see that , a contradiction. In the case, comparison of maximal terms again reveals that . Now
[TABLE]
We observe that , but is decomposable unless by Theorem 2.1. For , we deduce that is properly contained in . Thus we have , and , as required. ∎
We may conjugate (applying equation 1.1) to complete the case where is linear.
Corollary 3.3**.**
Let be partitions of respectively. We suppose that . If
[TABLE]
then either and or we are in the exceptional case
[TABLE]
Let be arbitrary partitions of respectively. We now consider what the condition
[TABLE]
tells us about this quadruple of partitions. We first suppose that . This implies that , say. Furthermore,
[TABLE]
We set , and set , , , . Since , we note that and so . Since and are coprime, as are and , it follows that and . Thus
[TABLE]
From equation 3.5, we observe that implies , and so we can set for all . Now, and so the final row length satisfies
[TABLE]
We have a partition with , and, in a similar fashion, we deduce that . Without loss of generality, we now assume that . We plug in our equalities and back into equation 3.5 and to show that
[TABLE]
We immediately obtain the following corollary.
Corollary 3.4**.**
Let be partitions of , respectively. We suppose that . If
[TABLE]
then and .
Proof.
By the discussion above, we know that we are dealing with a quadruple
[TABLE]
Comparing the width of the partitions on the left and right of
[TABLE]
we deduce that . Thus , , and thus , as required. ∎
We now consider the case where the lengths of the partitions and (and hence and ) differ. We suppose (without loss of generality) that . We set and for some . Thus , say. Observe that if and only if the partitions
[TABLE]
coincide. We deduce that
[TABLE]
for , and
[TABLE]
and, in order for to be a partition, we need
[TABLE]
which, rearranging, gives
[TABLE]
We are now ready to complete our proof of Theorem A.
Theorem 3.5**.**
Let be partitions of , respectively. We suppose that both and are non-linear and . If
[TABLE]
then and .
Proof.
We set and for . We first see what can be deduced from . From equation 3.6 we have that
[TABLE]
for and , and, from equation 3.7, we deduce that and so which implies . From equation 3.6 this implies that in other words .
We now see what can be deduced from . We have already concluded that . Therefore applying equation 3.6 (but with the partitions , , and ) we deduce that
[TABLE]
for some and .
From equation 3.8 and 3.9 we deduce that can be built from boxes of size . In other words,
[TABLE]
for some . Since might have repeated parts, we write in the form
[TABLE]
where , so
[TABLE]
Now, equation 3.8 reveals that
[TABLE]
where and, from equation 3.9,
[TABLE]
where . By looking at the first row of we deduce that provided the last part of is and that it appears with multiplicity . This implies that
[TABLE]
But the sum over these final rows is which implies and and that
[TABLE]
Now we input this into equation 3.10 to deduce that
[TABLE]
On the other hand by equation 3.11 we know that
[TABLE]
Therefore
[TABLE]
and thus or . If then , contrary to our earlier observation that . If , then , contrary to our assumption that is non-linear.
Finally, it remains to consider the case. This is the case in which is a rectangle. Here we have that and
[TABLE]
Now, recall that ; and so
[TABLE]
and so the rectangle in equation 3.12 is at least 2 rows shorter than that in equation 3.13. This implies that and or and so is linear, a contradiction. ∎
We have now classified all possible equalities between products where neither, one, or both of and are linear partitions. This completes the proof of Theorem A.
Acknowledgements**.**
We would like to thank Cedric Lecouvey for bringing to our attention the question of unique factorisability of products of Schur functions and in particular for introducing us to Rajan’s result for Littlewood–Richardson products.
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