Young flattenings of polynomials are equivariant linear maps from a space of homogeneous polynomials to a space of matrices, where the row and column space are irreducible representations of . One is usually interested in finding lower bounds for the rank of the image of a Young flattening, as it can be used to obtain lower bounds on the border Waring rank of a polynomial, and more generally for any border -rank for a -variety , i.e., given a point to find a lower bound on the smallest such that lies on the -th secant variety of , see e.g. [Lan15]. One early example are Sylvester’s catalecticants [Syl52]. Landsberg and Ottaviani [LO15]
use Young flattenings in the tensor setting. The nameYoung flattening was introduced in the predecessor paper [LO13]. Young flattenings also appear in disguise in the area of algebraic complexity theory as matrices of partial derivatives, shifted partial derivatives, evaluation dimension, and coefficient dimension [NW95, GKKS14]. They can in principle be used to find computational complexity lower bounds in many algebraic computational models such as border determinantal complexity (see [LMR13]) and border continuantal complexity [BIZ18], which makes Young flattenings an interesting tool in the Geometric Complexity Theory approach by Mulmuley and Sohoni [MS02], [MS08], [BLMW11]. Limits of these methods (in the case of studying -rank) have recently been proved in [EGOW18, GMOW19]. No such limits are known for using Young flattenings to study the orbit closure containment problems in geometric complexity theory. First results in this direction were obtained in [ELSW18], where limits to the method of shifted partial derivatives are shown. This was improved on in [GL19], where a setting was given in which Young flattenings give strictly more separation information than partial derivatives.
The Waring rank of a homogeneous degree polynomial is defined as the smallest such that can be written as a sum of many -th powers of homogeneous linear forms (arbitrary linear combinations of -th powers are usually allowed if the base field is not algebraically closed). For example , hence has Waring rank at most 2. The border Waring rank of is the smallest such that can be approximated arbitrarily closely coefficient-wise by polynomials of Waring rank at most . For example , hence has border Waring rank at most 2.
If a Young diagram is contained in another Young diagram such that the column lengths of both diagrams differ by at most 1 in each column, then we have a unique nonzero equivariant map between where , , and are irreducible polnomial -representations, and is the difference in the number of boxes of and . This is called the Pieri map, and it induces a linear map Since border Waring rank is subadditive, a lower bound on the border Waring rank of is obtained by rounding up to quotient of ranks
where is some variable that appears in , and rank is the usual rank of matrices.
There are several isomorphic constructions for the irreducible polynomial representations of the general linear group in characteristic zero. The two best known versions are called Schur modules and Weyl modules and they only differ in the order of the row-symmetrizer and the column-symmetrizer in their definition of the Young symmetrizer. This results in different bases for the irreducible representations. Sometimes results that are proved in one basis are reproved in the other basis, but the proofs look significantly different (see e.g. [BCI11] and [MM14]). In fact, so far some results are only provable in a natural way over one basis and not the other, see e.g. [Res20]. Based on an explicit paper by Olver over the Weyl module basis [Olv82] Steven Sam in 2009 implemented his Macaulay2 package PieriMaps [Sam08], which among other things can be used to compute the rank quotient (1.1), see Section 4 (A) below.
The papers [Far16] (in its Section 5111Although the description in the paper is wrong, the use of the software package is correct and gives the result claimed in the paper.) and [Oed16] (only in version 1) describe the PieriMaps package as if it would be working in the Schur module basis and they assume that the Young flattenings have an extremly simple combinatorial description. However, this is wrong (see Section 5 below), which led to a revision of [Oed16].
In this paper we work out the details of Young flattenings in the Schur module basis: We closely mimic the arguments in [Olv82], but we take care of subtle sign issues that are not present in Olver’s work over the Weyl module basis. We then make use of a recent fast algorithm (and implementation) by Reuven Hodges for Young tableau straightening in the Schur module basis [Hod17] to get a highly efficient Young flattening algorithm that outperforms Sam’s PieriMaps implementation by several orders of magnitude in many examples. We obtain the most impressive speedup factor of 1000 for flattening the power of a linear form, which is the denominator of (1.1).
Our contribution is therefore twofold: We thoroughly clarify the theory of Young flattenings in the Schur module basis and we present a new and efficient implementation for Young flattenings that uses Hodges’ state-of-the-art straightening algorithm over the Schur module basis.
A composition of a number is a finite list of natural numbers adding up to , i.e., is a composition of 9. A partition is a nonincreasing composition, for example is a partition. We write if is a partition of . We write if is greater than the number of entries in . We define . We identify a partition with its Young diagram, which is a top-left justified array of boxes, i.e., the set of points . For example, the Young diagram corresponding to is
and we have and . We see that is the number of rows of the Young diagram corresponding to . We denote by the number of boxes in , i.e., . We denote by the Young diagram obtained by reflecting at the main diagonal, e.g., . It follows that is the length of the -th column of . We write if for all we have . If , then we denote by the set of points that are in but not in . We call a horizontal strip if it has at most 1 box in each column. In this situation we write .
A Young diagram whose entries are labeled with numbers is called a Young tableau of shape . For example,
is a Young tableau of shape . A Young tableau is called semistandard if the entries strictly increase in each column from top to bottom and do not decrease in each row from left to right. For example, centertableaux,notabloids 1113,23 is a semistandard tableau. We denote by the symmetric group on the set of positions in . The group acts on the set of all Young tableaux of shape by permuting the positions. We write for the permuted Young tableau, where is the shape of and . For a subset of positions we write to denote the symmetric group that permutes only the positions in among each other and fixes all other positions.
Let be the
-th tensor power of a vector spaceand associate to every tensor factor a position in . A rank 1 tensor can now be represented by a Young diagram in whose -th box we write the vector . If we fix a basis of , then a basis of is obtained by all ways of writing into the boxes of , allowing repetitions. If the fixed basis is clear from the context, then we write instead of into the boxes and obtain a Young tableau. The basis vector corresponding to the Young tableau is also denoted by when no confusion can arise, so for example if we can use the multilinearity of the tensor product to write
2 (A) The Weyl module basis
Let λ be a Young diagram and let such that (i.e., the box below is still in ). Then we define Pictorially, is the subset of the boxes of given by collecting all boxes on the following path: Start at position and move from left to right along row to box , then switch the row to and move along row until reaching . For example, is given by the dotted boxes in the following diagram:
2.1 Definition (Weyl module, [Wey03, 2.1.15]).
The Weyl module is defined as the quotient space , where is the linear subspace generated by the following two types of vectors:
(Symmetric relation) , if σ is a permutation that preserves the row indices of all positions of (in other words, permutes within the rows of ).
(Shuffle relation) if such that .
The Weyl modules for Young diagrams with at most rows form a complete list of pairwise non-isomorphic irreducible polynomial representations of . In this paper we will not work with Weyl modules, but with the isomorphic Schur modules, which are defined in the following section.
2 (B) The Schur module basis
Let be a Young tableau of shape . Let be two column indices. Let and be two equally large sets of boxes, from column and from column . An exchange tableau of corresponding to and is defined as the tableau arising from by exchanging the content of the boxes with the content of the boxes while preserving the vertical order of the entries in and . We denote this exchange tableau by . For a subset of boxes from column , we write , where ranges over all cardinality subsets of boxes in column .
2.2 Definition (Schur module).
The Schur module is defined as the quotient space where is the linear subspace generated by the following vectors
(Grassmann relation) , where is obtained from by swapping two elements in the same column.
(Plücker relation) for any and any subset of a column with .
For example, in we have and we have by the Grassmann relation. The Plücker relation gives
via . Note that is the set containing the single box in row 1 and column 2.
2 (C) The Schur module via moving boxes between columns
If we only consider the Grassmann relation, then we call the quotient . More formally, let be the linear subspace spanned by the , where is obtained from by swapping two elements in the same column. The quotient is denoted by
. Clearly, in the language of skew-symmetric powers we have
In terms of explicit basees, this isomorphism maps each column from top to bottom to a skew-symmetric tensor and vice versa: For example, is mapped to . We define
as an outer direct sum. Note that each , but there is no obvious embedding of in . In fact, there is a natural isomorphism
that can be described explicitly in terms of basis vectors using first the isomorphism (2.3): A standard basis vector
is mapped to
for example is mapped to . We have an action of on , which induces an action of via the group homomorphism , i.e., sending matrices to block diagonal matrices that have many copies of on their main diagonal. With this action of , the isomorphism in (2.4) is an isomorphism of -representations.
For given , , we consider the Lie algebra element that is a block matrix as follows: the block matrix is zero everywhere but the block
is the identity matrix on. Clearly applying is a -equivariant map. Acting with on the tensor from (2.5) gives
Here for a tensor we define as the tensor obtained by “removing the -th tensor position” (this is only well-defined if a basis of is fixed and an ordering of the basis vectors is fixed). For example, if , then , and .
Note that if , then . Define . Note that if , then , hence
2.8 Proposition ([Tow79, Cor. 1]).
and are isomorphic representations of . The isomorphism maps each basis vector given by a semistandard tableau to a basis vector corresponding to the same semistandard tableau.
2.9 Proposition ([Tow77, Thm. 2.5]).
and are isomorphic representations of .
3 Young Flattenings in the Schur module basis
We give an explicit description of the construction of the so-called Pieri inclusions defined on the basis of Schur modules. Olver [Olv82] first described the corresponding construction based on Weyl modules and we closely mimic this construction while taking care of the subtle signs that are introduced when using the Schur module basis. To the best of our knowledge, this construction has never been explicitly described for Schur modules. This algorithm will directly give the construction for Young flattenings.
centertableaux, mathmode, boxsize=3em
mathmode, boxsize=1em, centertableaux, smalltableaux
Pieri’s well-known formula states the following isomorphism of -representations:
The resulting -equivariant inclusions
are called Pieri inclusions and are unique up to scale by Schur’s lemma. We define by composing Pieri inclusions for as follows. For let be the sequence of partitions obtained by adding one box at a time to from left to right, so that after having added boxes we arrive at . Then define the -equivariant map via
By proving that the restriction of to is nonzero (see Lemma 3.10), it immediately follows that this restriction equals (up to a nonzero scalar). It remains to describe for which has only a single box and then prove Lemma 3.10.
3 (A) The single box case
We assume that consists of a single box.
We will define the linear map , show its -equivariance (Lemma 3.4), and then prove that it is well-defined on the quotient space if we interpret its image in the quotient space (Theorem 3.5). In this way, induces a map . By the uniqueness of the Pieri inclusions we have that this map equals or is the zero map (nonzeroness is proved in Lemma 3.10). Note that by definition.
Let be a finite sequence of positive integers. Define the linear map
Pictorially, this means that a box is shifted from column to , then from to , and so on. Clearly is -equivariant as the composition of -equivariant maps. Note that if and moves a box from an empty column to another column, then .
We consider the set of all strongly decreasing sequences of natural numbers from to , which we denote by
For let be the hook length in of the box in column and row . Let be the column where and differ. We define as the product of all hook lengths in with respect to column , i.e.,
Finally, we define by
The map maps into .
is a linear combination of -equivariant maps. ∎
Let be partitions with being obtained from λ by appending a single box in row . Then and hence is well-defined.
Before proving Theorem 3.5 we first have to prove the following lemma.
We denote by the commutator of two linear maps and .
Let for some column lengths with and and let . Then,
Note the similarity to Lemma 5.4 in [Olv82] with the exception that the sign in the first case is reversed. Moreover, [Olv82] ignores handling the special case when column lengths vanish. We handle these cases explicitly. If , and , then
Moreover, if , , and , then
Proof of Lemma 3.6.
We focus on the key positions in the tensor.
: , .
: , , .
: Equivalent to the case but changing the order of elements in the commutator. Thus, the sign changes:
: If all are pairwise distinct, then both maps affect distinct columns and hence they commute. We first treat the case , .
We now treat the remaining case , . For a basis vector let denote the basis vector with positions and removed.
where is 1 if and 0 otherwise. Here we used the notation
Note that the second case happens exactly when .∎
We will make heavy use of the following identity: Let be linear maps, then
The rule can be interpreted as the Leibniz rule for .