BLOCK STANLEY DECOMPOSITIONS
I. ELEMENTARY AND GNOMON DECOMPOSITIONS
Department of Mathematics
Iowa State University
Ames, Iowa 50011
823 Carroll Ave.
Ames, Iowa 50010
This is the final preprint. The published version appears in J. of Pure and Applied Algebra 219 (2015) 2189-2205. The doi is 10.1016/j.jpaa.2014.07.030.
Running head: BLOCK STANLEY DECOMPOSITIONS I
ABSTRACT: Stanley decompositions are used in invariant theory and the theory of normal forms for dynamical systems to provide a unique way of writing each invariant as a polynomial in the Hilbert basis elements. Since the required Stanley decompositions can be very long, we introduce a more concise notation called a block decomposition, along with three notions of shortness (incompressibility, minimality of Stanley spaces, and minimality of blocks) for block decompositions. We give two algorithms that generate different block decompositions, which we call elementary and gnomon decompositions, and give examples. Soleyman-Jahan’s criterion for a Stanley decomposition to come from a prime filtration is reformulated to apply to block decompositions. We simplify his proof, and apply the theorem to show that elementary and gnomon decompositions come from “subprime” filtrations. In a sequel to this paper we will introduce two additional algorithms that generate block decompositions that may not always be subprime, but are always incompressible.
Let be a monomial ideal and let be the set of standard monomials, that is, the monomials that are not in . Each coset in has exactly one representative as a standard polynomial in , and is isomorphic to as a vector space over , although the ring structure of is lost. A block decomposition of (defined precisely in Section 2) is an expression for as a disjoint union of rectangular blocks of standard monomials, regarded as points in Newton space. A block will be called a Stanley block if its span is a Stanley space; a block decomposition of into Stanley blocks is equivalent to a Stanley decomposition of .
In this paper and its sequel we study algorithms leading from to a block decomposition for , emphasizing that different algorithms produce block decompositions with different algebraic and geometric properties. Our long-range goal is to produce block decompositions that are as simple as possible, with respect to various criteria. In this first paper we present two algorithms, producing what we call the elementary and gnomon block decompositions respectively, and show that these are associated with subprime filtrations of , a natural generalization of prime filtrations. In doing so, we generalize Soleyman-Jahan’s criterion in [16, Prop. 2.7] for a Stanley decomposition to be associated with a prime filtration, and simplify the proof. The gnomon decomposition is shorter (contains fewer blocks) than the elementary decomposition. In the second paper we present algorithms leading to what we call the organized and stacked block decompositions. These are incompressible, in the sense that they cannot be shortened by combining blocks. We have not yet found an algorithm that always produces a minimal block decomposition (one with the fewest blocks), but since any such block decomposition must be incompressible, these algorithms are a step in that direction.
From one point of view, block decompositions are a generalization of Stanley decompositions, and are in general coarser than Stanley decompositions. From another point of view, each block decomposition has a unique minimal Stanley refinement, given by Algorithm 2.2 below, and since this algorithm is very easy, the block decomposition can be taken as a compressed notation for this Stanley decomposition. From this viewpoint, we are studying algorithms leading from to a Stanley decomposition of . The usual algorithm of this type, [17, Lemma 2.4], proceeds by induction on the variables . Our gnomon algorithm works by induction on the generators of , so that if one has determined a block decomposition for and then adds a generator to , it is not necessary to start over from the beginning.
In Section 2 we define block decompositions and Stanley decompositions and present Algorithm 2.2 that relates them. Sections 3 and 4 contain the algorithms for the elementary and gnomon block decompositions. Section 5 contains the generalization of Soleyman-Jahan’s theorem and the proof that elementary and gnomon decompositions are subprime.
Stanley decompositions are best known among algebraists in connection with the conjecture that the Stanley depth of a module over a polynomial algebra is an upper bound for the classical depth, but have a separate life in applied mathematics through their use in describing subalgebras. This life is based entirely on the first two pages of  and, for one application, on . The motivation for our work comes from that direction. Let be a subalgebra of . It is desired to find a formula, containing finitely many arbitrary polynomial functions, that expresses each element of exactly once as the arbitrary functions are varied. A set of polynomials is a Hilbert basis for if the monomials span as a vector space over . The expression for a given in terms of the Hilbert basis is usually not unique, because the monomials may not be linearly independent. There always exist linearly independent subsets of these monomials, which we call preferred sets, and the expression of any as a linear combination of these is unique. But a preferred set of monomials is infinite, and may not have a finite description leading to an expression involving arbitrary polynomial functions. Stanley decompositions in the manner of  provide suitable preferred sets . Let be the unique algebra homomorphism such that ; then is the ideal of relations among the . Compute a Gröbner basis for using an appropriate elimination order on ([1, p. 81]), and obtain the monomial ideal of leading terms of , which is generated by the leading terms of the Gröbner basis. The standard monomials of provide a preferred set of monomials in when each is replaced by . In fact is isomorphic as an algebra to , and isomorphic as a vector space to both and . Furthermore, a Stanley decomposition for provides the desired formula expressing each element of uniquely. For instance, if has a Hilbert basis of four elements with the single relation , and is taken as the leading term, then and the Stanley decomposition given by the algorithm of  is
It follows that every element of can be written uniquely in the form
where and are arbitrary polynomials in three variables. (This example arises for the classical seminvariants of the binary cubic form.)
Notice that the number of arbitrary polynomials is the same as the number of Stanley spaces. The number of such polynomials can become large in larger problems, and it is natural to ask if this can be reduced. In part this can be done by a good choice of the term order; for instance, if was taken as the leading term in the example, there would be three Stanley spaces instead of two. But even once is selected, there can be large differences in the number of Stanley terms in different Stanley decompositions for . This suggests the problem: Find a minimal Stanley decomposition for given a fixed monomial ideal . A related, but different, problem is to find a minimal block decomposition for given ; here the minimality is with regard to the number of blocks, not the number of Stanley spaces.
It is not an accident that the example mentioned above comes from classical invariant theory. Invariant theory is closely linked with normal form theory for systems of differential equations with nilpotent linear part. There was a flurry of work on this topic in the late 1980s and early 1990s (, , , , , ), and a second flurry more recently (, , [14, Ch.12], , , , , ). Additional motivation for our questions comes from these papers. The algebra of scalar fields that are invariant under the the group , where is a nilpotent matrix, plays an important role. These invariants, in turn, are the same as classical seminvariants of a binary form, or joint seminvariants of several forms. Knowing the seminvariants of several forms, transvectants are used to find a Hilbert basis for the the joint seminvariants. The box product method introduced in  lifts this procedure to the level of Stanley decompositions. The box product of two Stanley decompositions for the seminvariants of two binary forms is a Stanley decomposition for the joint seminvariants of the two forms; thus the uniqueness issue is taken care of automatically, without further Gröbner basis work. In [MurdBox] I (J.M.) have reformulated the box product as a product of block decompositions, in which distributes over disjoint union; the basic unit of computation is the box product of two blocks, for which algorithms are given, and the final result (expressed in Stanley decompositions) is shorter than for the method in . It is best, then, to start with block decompositions with a minimal number of blocks.
This research began when I (J.M.) asked T.M., a software engineer, to write programs for me concerning Stanley decompositions. He created block decompositions as a computer-friendly shorthand for Stanley decompositions, and the gnomon algorithm to provide an interactive program in which ideal generators could be added one at a time. I provided the mathematical write-up, proofs, and examples, and the material on subprime filtrations. (The word subprime was provided by the mortgage crisis.)
2. Block decompositions
Let be a vector subspace; is a monomial space if it is spanned by the set of monomials in , which is then also the unique monomial basis for . Under the one-to-one correspondence , between monomial spaces and their monomial bases, a direct sum of two spaces corresponds to the disjoint union of their bases. We often identify with in the Newton space .
A Stanley space is a monomial space of the form
where is a subset of the set of variables and is a monomial. An expression exhibiting a monomial space as a direct sum of finitely many Stanley spaces is called a Stanley decomposition of . (Because of this finiteness requirement, there are monomial spaces, such as , that do not have Stanley decompositions.)
Let be a monomial ideal in and let be the set of standard monomials for . Then is the space of standard polynomials, and as noted in the introduction, as vector spaces. If and , a Stanley decomposition for is
Applying the correspondences above, we replace each Stanley space by its set of monomials and replace by the disjoint union symbol , obtaining
In this way the Stanley decomposition (2.1) becomes the following block decomposition in Newton space:
More generally, let denote an integer interval, with the understanding that does not include and that if . Define a block in to be a Cartesian product of intervals:
If any , the block is empty. The bottom row of a block is called its inner corner. The letter will be used ambiguously to mean the matrix as such, the set of integer vectors represented by the matrix, and the set of monomials having those integer vectors as exponents. Let . A block decomposition of is an expression
where each is a block. For example, the Stanley decomposition (2.1) can be written as the block decomposition
corresponding to the diagram shown above in Newton space.
After this research was mostly completed, we learned of the interval decompositions defined in . They define the interval to be the set of monomials lying between and in division partial order. This is essentially the same as our block in (2.4), except that they consider only monomials that divide a chosen , so that, in particular, is not allowed in the top row. This is sufficient since they only consider interval decompositions for , where and are two monomial ideals in .
A block decomposition is called compressible if the union of some subset of the blocks is itself a block. In this case the decomposition can be simplified by performing the union. However, performing one such union may prevent another one, so a given decomposition may be compressible in several ways to give distinct incompressible decompositions. A decomposition is called minimal if there is no decomposition (describing the same set of monomials) having fewer blocks. Incompressibility and minimality are two possible notions of “simplicity” of a decomposition (as briefly discussed in the introduction). An incompressible decomposition need not be minimal, and a minimal decomposition need not be unique. The following example shows that incompressibility is a global property of a decomposition, and cannot be detected by examining the decomposition locally: The disjoint union
is compressible to the single block , although no proper subset of the five blocks can be compressed.
A Stanley block is a block (2.2) in which each equals either or . The span of a Stanley block is a Stanley space, and any Stanley decomposition can be changed to a block decomposition by replacing Stanley spaces by Stanley blocks and by . Such a block decomposition can almost always be compressed. For example, becomes
Any block can be written as a disjoint union of its bounded part and its unbounded part; the bounded part is obtained by replacing the unbounded columns with zeroes, the unbounded part by doing the same with the bounded columns. For instance,
Let be the set of variables associated with unbounded columns of ( and in the example). Let be the monomial in the bottom row of the unbounded part ( in the example). Let be the finite set of monomials in the bounded part (, , , , , , with ). Let for . Then the block is converted to its minimal Stanley decomposition as follows.
The Stanley decomposition with the fewest Stanley spaces that describes the span of the monomials in is
Note that all of the coefficient algebras in the minimal Stanley decomposition of a block are equal, and that the number of Stanley spaces is the product of the numbers of elements in the intervals defined by the bounded columns of . When Algorithm 2.2 is applied to each block in a block decomposition, the coefficient rings may differ from block to block, but the total number of Stanley spaces is just the sum of the number for each block.
As an introduction to the next two sections, we now give examples of the elementary and gnomon block decompositions defined in those sections. For
the elementary decomposition of is the disjoint union of the blocks
The gnomon decomposition comes in two forms,
The elementary decomposition is compressible to either of the gnomon decompositions, and (in this case, but not always) the gnomon decompositions are incompressible.
3. The elementary block decomposition
In Newton space, the division partial order will be written as , meaning for . For variables in Newton space we use , reserving for constants, so that an equation such as will represent the “hyperplane” through perpendicular to the axis in . Let and let be the monomial ideal with the indicated generators; we also write . It is assumed that these generators are minimal, that is, no redundant generators (divisible by another generator) are included. The elementary block decomposition for is created by first gridding the Newton space with the hyperplanes for and , passing through all points of the minimal generating set, and then discarding those of the resulting blocks that belong to rather than . The following figure illustrates the elementary decomposition (2.6) at the end of Section 2. The solid dots are the generators at and , the dark lines are the boundary of , and the blocks are numbered in the order that they appear in (2.6).
The elementary block decomposition is almost always compressible, and so is not very desirable in itself, but it shows that a natural block decomposition always exists. The other decompositions (gnomon, organized, and stacked) in this paper and its sequel can be obtained by compressing the elementary decomposition, so we will never need to consider blocks smaller than the blocks in the elementary decomposition, or blocks that are not disjoint unions of elementary blocks.
The following is a precise algorithm to create the elementary decomposition.
Let be presented by its minimal generating set. The elementary block decomposition is generated as follows.
For , create the list of exponents of in the set of generators, adding zero at the beginning of each list:
Delete any repetitions in each list , and re-order each list in increasing numerical order.
Create a preliminary list of inner corners as follows. For , choose an entry from ; then create an inner corner . Do this in all possible ways.
Refine the list of inner corners by discarding any , that is, any such that for some .
Create a block for each inner corner in the refined list, by determining its outer corner , as follows. If has a successor in (under the ordering from step 2), put . If is the last element of , but .
Let be the number of resulting blocks, and enumerate these as
For the example (2.5), the lists from Step 2 are , . The preliminary list of inner corners (in lexicographic order) is . From this we discard , , and . It is now easy to check that the outer corners in (2.6) follow from step 5 above.
The elementary decomposition has the following property, which will be used in Section 5. Let be a monomial. Then there exists a unique inner corner of the elementary decomposition for that is maximal (under ) among all inner corners satisfying . The inner corner satisfying this condition determines the block that contains . (The proof is trivial: The block containing has for the -th component of its inner corner the largest element of that is .)
4. The gnomon decomposition
Consider the following Block Subtraction Problem: Given a block (2.2) and a principal monomial ideal , find a block decomposition of . If is within the block , there are distinct natural solutions. For , the following figure shows a block in , an ideal generated by the heavy dot, and the two block decompositions of . The set of points in is the sort of L-shaped region the Pythagoreans called a gnomon ().
The first of these decompositions, expressed algebraically, takes the form
We will generalize this formula to arbitrary in such a way that it gives all solutions, one for each permutation of the variables . First we focus on the natural order of variables (the trivial permutation), which for gives (4.1).
The following solution for is valid whether or not is in the interior of , and reduces to (4.1) when it is:
The first block consists of all monomials in that are not divisible by , and is empty if . The second block consists of those monomials in that are divisible by but not by . But this solution has a defect: it sometimes divides unnecessarily into two nonempty blocks whose union is again . This happens whenever but . In fact, if for 1 or 2 or both, then and . This observation also simplifies the formula in the remaining cases (where for all ), because then can be replaced by .
Generalizing to higher dimensions, we define the gnomon determined by and to be the set represented by the matrix
as follows. (Placing between and is intended to suggest “cutting by ”.)
If there exists such that (that is, if ), then
If for all , (), let
The following lemma shows that the solution to the Block Subtraction Problem is a gnomon as defined above.
For all nonnegative values of , , and ,
Next assume that for all , so that . Then the original block can be decomposed into two subblocks by the hyperplane :
(Note that occurs in the lower left corner of the second matrix where is expected, but these are equal.) The first of these subblocks lies outside of , and belongs to the difference we are computing; it equals the first block of (4.4). The second subblock is carried forward to the next step in a recursive process: It is split by the hyperplane into two subblocks, one having and the other having . Again, the first subblock is retained and the second carried forward to the third step, and so forth.
Finally, if , then and the first subblock in (4) is empty. Furthermore the lower left entry in the second matrix should be (rather than , which would cause the block to include monomials that are not in the original block). But in this case , so (4) remains correct. The same reasoning applies if the condition is encountered later in the recursion. ∎
Now let be a monomial ideal for which a block decomposition is known, and let be a principal monomial ideal. Let be the sum of these ideals (in the usual sense), and note that We refer to this as adding a generator to .
A block decomposition for is given by
The standard monomials of are the monomials that are standard for and, in addition, are not divisible by . We remove the monomials divisible by from each block of standard monomials for by finding the gnomons . Since these are disjoint, their union is a block decomposition.∎
Here are the terms of in the order shown in equation (4.4), retaining any empty blocks, or, if (4.2) applies, then and for . Retaining the empty sets allows a uniform notation in the proof of Theorem 5.5.
Let be a monomial ideal given by its minimal generators. The gnomon decomposition of , with respect to the standard order of the variables, is defined to be the result of applying Algorithm (4.4) below. The result may depend on the order in which the generators are indexed; this noncommutativity of the generators is illustrated in the examples below.
Apply Algorithm 4.2 repeatedly, beginning with and adding the generators in that order.
Let be a permutation of the integers , regarded as specifying an order of the variables. The -gnomon decomposition of is defined by changing variables temporarily to , applying algorithm (4.4) using the new variables, and then returning to the original variables. (The generators are still taken in their indicated order.)
The blocks of the gnomon decomposition are unions of blocks of the elementary decomposition, because whenever a block is subdivided in Algorithm 4.4, the subdivision occurs along a portion of a hyperplane for some and . The subdivisions for elementary blocks occur along these same hyperplanes (but along the entire hyperplane, not a portion of it.)
The first block contains the monomials that are not divisible by , the second those that are divisible by but not by . Next we find
since and is not in the block. Finally
The result of these calculations is (2.7).
To show noncommutativity of generators, the gnomon decomposition of is
which is incompressible; the opposite order of generators gives instead
This is compressible (to the previous result), since
We conclude this section with two lemmas that will be used in Section 5.
With as in (4.3),
Compute the unions successively starting from the right and working to the left, noticing that each union is between matrices that differ only in one column. For instance, if the first step is
If the result is clear; if then the left-hand block is empty and again the result is clear. (The lemma and proof remain valid without assuming , but in that case it does not apply to the gnomon decomposition, which is governed instead by (4.2).) ∎
5. Subprime filtrations
A filtration of by monomial ideals is a finite nested sequence
This is often called a filtration of , and it does induce an actual filtration of by subrings, namely . If each set difference is a block , we call (5.1) a subprime filtration. If, further, each block is a Stanley block, we call (5.1) a prime filtration; this is simpler than the usual definition in the literature, but is equivalent, as pointed out at the end of this section. A block decomposition, with the blocks indexed in a specific order, will be called subprime (or prime) if it is associated in this way with a subprime (or prime) filtration. In this section all block decompositions are taken as ordered block decompositions. This simplifies the discussion, in that we need not consider all possible orderings before deciding whether a given block decomposition is subprime.
The following equations hold trivially for subprime filtrations. Quotients, direct sums, and isomorphisms are to be understood as quotients, direct sums, and isomorphisms of vector spaces, and is the set of standard monomials for in .
That is, with each subprime filtration of is associated a block decomposition in which the blocks are naturally ordered so that the -th block , and when the block spaces are successively added (by direct sum) to the (considered as a vector space), the ideals of the filtration are reconstructed. Thus the subprime filtration and the (ordered) block decomposition contain the same information. Prime filtrations and (ordered) Stanley decompositions are related in the same way.
A convenient notation to exhibit the relationship between ideals and blocks in a subprime filtration is
For example, the block decomposition (2.4) is associated with a subprime filtration as follows:
with , , and .
Observe that the ideals in the filtration can be found by adding the inner corner of each block to the generators of the previous ideal. For instance, adding the inner corner of to gives . See Lemma 5.3 below.
It is known that there exist Stanley decompositions that are not associated in this way with prime filtrations; thus there are also block decompositions that are not associated with subprime decompositions. The standard example, introduced in , is , ,
In this example, the Stanley blocks cannot be ordered in such a way that when added successively to , they produce monomial ideals. Here is our version of Soleyman-Jahan’s theorem characterizing Stanley decompositions that come from prime filtrations.
An ordered block decomposition for is subprime if and only if is a monomial ideal for each .
The direction has already been proved. For the direction, the hypothesis implies that is a vector space basis for an ideal for each , and that these ideals form a filtration. It is equally clear that each , so the filtration is subprime.∎
Note that when written as a block decomposition, the blocks of (5.7) are unions of blocks from the elementary decomposition of , just as is the gnomon decomposition. This shows that although the elementary decomposition itself is subprime (Theorem 5.4 below), the result of compressing an elementary decomposition is not in general subprime. But the gnomon decomposition, which is a compression of the elementary decomposition, is subprime (Theorem 5.5).
If is any block (2.2), and , we define the th outer adjacent face of to be the block
(This is not a subset of , but is displaced by one step in the direction from the ordinary th outer face of .) If we set .
Let be a block with inner corner , and let be a monomial ideal disjoint from . Let . Then spans a monomial ideal (or equivalently, for some monomial ideal ) if and only if the outer adjacent faces of are contained in . In this case and .
A set of monomials spans an ideal if and only if every monomial divisible by () an element of belongs to . Let and suppose that spans a monomial ideal. Let for some . Then , so . But , so . Therefore for each . Conversely, suppose that for each . Since every element of that is not in is divisible by some element of , and therefore belongs to , we conclude that . Therefore , so spans an ideal. ∎
We are now ready to prove that elementary and gnomon decompositions (with suitable orderings) are subprime.
Let be a monomial ideal and let be the elementary block decomposition of , ordered according to Remark 3.2. Let for ; that is, reverse the ordering of the blocks. Then is a subprime decomposition.
We work in the superscript notation, restating the criterion of Theorem 5.2 as
The proof begins with and works backward. Let be the refined list of inner corners of elementary blocks for , created in step 4 of Algorithm 3.1, in lexicographic order according to Remark 3.2. Since implies , is a maximal element of under .
We claim that each outer adjacent face is contained in . Let . Then and