Positroid varieties: juggling and geometry
While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, the intersection of only the cyclic shifts of one Bruhat decomposition turns out to have many of the good properties of the Bruhat and Richardson decompositions.
This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown-Goodearl-Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties.
Applying results from [KnLamSp], we show that positroid varieties are normal, Cohen-Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and Buch-Kresch-Tamvakis’ approaches to quantum Schubert calculus.
- 1 Introduction, and statement of results
- 2 Some combinatorial background
- 3 Affine permutations, juggling patterns and positroids
- 4 Background on Schubert and Richardson varieties
- 5 Positroid varieties
- 6 Examples of positroid varieties
- 7 The cohomology class of a positroid variety
- 8 Quantum cohomology, toric Schur functions, and positroids
1. Introduction, and statement of results
1.1. Some decompositions of the Grassmannian
This paper is concerned with the geometric properties of a stratification of the Grassmannian studied in [Lus98, Pos, Rie06, BroGooYa06, Wi07]. It fits into a family of successively finer decompositions:
We discuss the three known ones in turn, and then see how the family of positroid varieties fits in between.
The Bruhat decomposition of the Grassmannian of -planes in -space dates back, despite the name, to Schubert in the 19th century. It has many wonderful properties:
the strata are easily indexed (by partitions in a box)
it is a stratification: the closure (a Schubert variety) of one open stratum is a union of others
each stratum is smooth and irreducible (in fact a cell)
although the closures of the strata are (usually) singular, they are not too bad: they are normal and Cohen-Macaulay, and even have rational singularities.
The Bruhat decomposition is defined relative to a choice of coordinate flag, essentially an ordering on the basis elements of -space. The Richardson decomposition is the common refinement of the Bruhat decomposition and the opposite Bruhat decomposition, using the opposite order on the basis. Again, many excellent properties hold for this finer decomposition:
it is easy to describe the nonempty intersections of Bruhat and opposite Bruhat strata (they correspond to nested pairs of partitions)
it is a stratification, each open stratum is smooth and irreducible, and their closures are normal and Cohen-Macaulay with rational singularities [Bri02].
At this point one might imagine intersecting the Bruhat decompositions relative to all the coordinate flags, so as not to prejudice one over another. This gives the GGMS decomposition of the Grassmannian [GeGorMacSe87], and as it turns out, these good intentions pave the road to Hell:
This raises the question: can one intersect more than two permuted Bruhat decompositions, keeping the good properties of the Bruhat and Richardson decompositions, without falling into the GGMS abyss?
The answer is yes: we will intersect the cyclic permutations of the Bruhat decomposition. That is to say, we will define an open positroid variety to be an intersection of Schubert cells, taken with respect to the the cyclic rotations of the standard flag. We will define a positroid variety to be the closure of an open positroid variety. See section 5 for details.
It is easy to show, though not immediately obvious, that this refines the Richardson decomposition. It is even less obvious, though also true, that the open positroid varieties are smooth and irreducible (as we discuss in Section 5.4).
There is a similar decomposition for any partial flag manifold , the projection of the Richardson stratification from . That decomposition arises in the study of several seemingly independent structures:
the characteristic notion of Frobenius splitting ([KnLamSp]).
We show that the positroid stratification and the projected Richardson stratification coincide. Specifically, we prove:
Theorem (Theorem 5.9).
If is a Richardson variety in the full flag manifold (), then its image under projection to is a positroid variety. If is required to be a Grassmannian permutation, then every positroid variety arises uniquely this way.
Theorem 5.9 has been suspected, but has not previously been proved in print, and is surprisingly difficult in its details. This result was already known on the positive part of , as we explain in Remark 1.2.
Once we know that positroid varieties are projected Richardson varieties, the following geometric properties follow from the results of [KnLamSp]. Part (1) of the following Theorem was also established by Billey and Coskun [BiCo] for projected Richardson varieties.
Positroid varieties are normal and Cohen-Macaulay, with rational singularities.
Though positroid varieties are defined as the closure of the intersection of cyclically permuted Bruhat cells, they can also be defined (even as schemes) as the intersection of the cyclically permuted Schubert varieties. In particular, each positroid variety is defined as a scheme by the vanishing of some Plücker coordinates.
The standard Frobenius spliting on the Grassmannian compatibly splits all positroid varieties. Furthermore, positroid varieties are exactly the compatibly split subvarieties of the Grassmannian.
Before going on, we mention a very general construction given two decompositions , of a scheme , one refining the other. Assume that
for each , there exists a subset such that ,
each is irreducible (hence nonempty), and each is nonempty. (We do not assume that each is irreducible.)
Then there is a natural surjection taking to the unique such that , and a natural inclusion taking to the unique such that is open in . (Moreover, the composite is the identity.) We will call the map the -envelope, and will generally use the inclusion to identify with its image. Post this identification, each corresponds to two strata , , and we emphasize that these are usually not equal; rather, one only knows that contains densely.
To each GGMS stratum , one standardly associates the set of coordinate -planes that are elements of , called the matroid of . (While “matroid” has many simple definitions, this is not one of them; only realizable matroids arise this way, and characterizing them is essentially out of reach [Va78].) It is a standard, and easy, fact that the matroid characterizes the stratum, so via the yoga above, we can index the strata in the Schubert, Richardson, and positroid decompositions by special classes of matroids. Schubert matroids have been rediscovered many times in the matroid literature (and renamed each time; see [BoDM06]). Richardson matroids are known as lattice path matroids [BoDM06]. The matroids associated to the open positroid varieties are exactly the positroids [Pos] (though Postnikov’s original definition was different, and we give it in the next section).
In our context, the observation two paragraphs above says that if a matroid is a positroid, then the positroid stratum of is usually not the GGMS stratum of , but only contains it densely.
For each positroid , Postnikov gives many parametrizations by of the totally nonnegative part (whose definition we will recall in the next section) of the GGMS stratum of . Each parametrization extends to a rational map ; if we use the parametrization coming (in Postnikov’s terminology) from the Le-diagram of then this map is well defined on all of . The image of this map is neither the GGMS stratum nor the positroid stratum of (although the nonnegative parts of all three coincide). For example, if and is the “uniform” matroid in which any two elements of are independent, this parametrization is
The image of this map is the open set where , , , and are nonzero. It is smaller than the positroid stratum, where can be zero. The image is larger than the GGMS stratum, where is also nonzero.
One may regard this, perhaps, as evidence that matroids are a philosophically incorrect way to index the strata. We shall see another piece of evidence in Remark 5.17.
1.2. Juggling patterns, affine Bruhat order, and total nonnegativity
We now give a lowbrow description of the decomposition we are studying, from which we will see a natural indexing of the strata.
Start with a matrix of rank (), and think of it as a list of column vectors . Extend this to an infinite but repeating list where if . Then define a function by
Since , each , and each with equality only if . It is fun to prove that must be , and has enough finiteness to then necessarily be onto as well. Permutations of satisfying are called affine permutations, and the group thereof can be identified with the affine Weyl group of (see e.g. [EhRe96]).
This association of an affine permutation to each matrix of rank depends only on the -plane spanned by the rows, and so descends to , where it provides a complete combinatorial invariant of the strata in the cyclic Bruhat decomposition.
This map from the set of positroid strata to the affine Weyl group is order-preserving, with respect to the closure order on positroid strata (Postnikov’s cyclic Bruhat order) and the affine Bruhat order, and identifies the set of positroids with a downward Bruhat order ideal.
Consequently, the cyclic Bruhat order is Eulerian and EL-shellable (as shown by hand already in [Wi07]).
We interpret these physically as follows. Consider a juggler who is juggling balls, one throw every second, doing a pattern of period . At time , they throw a ball that lands shortly222almost exactly at time , according to video analysis of competent jugglers before, to be thrown again at, time . No two balls land at the same time, and there is always a ball available for the next throw. If we let be the throw at time , this cyclic list of numbers is a juggling pattern333Not every juggling pattern arises this way; the patterns that arise from matrices can only have throws of height . This bound is very unnatural from the juggling point of view, as it excludes the standard -ball cascade with period . or siteswap (for which our references are [Pol03, Kn93]; see also [BuEiGraWr94, EhRe96, War05, ChGra07, ChGra08]). This mathematical model of juggling was developed by several groups of jugglers independently in 1985, and is of great practical use in the juggling community.
If is generic, then the pattern is the lowest-energy pattern, where every throw is a -throw.444These juggling patterns are called “cascades” for odd and “(asynchronous) fountains” for even. At the opposite extreme, imagine that only has entries in some columns. Then of the throws are -throws, and are -throws.555These are not the most excited -ball patterns of length ; those would each have a single -throw, all the others being -throws. But juggling patterns associated to matrices must have each .
If one changes the cyclic action slightly, by moving the first column to the end and multiplying it by , then one preserves the set of real matrices for which every submatrix has nonnegative determinant. This, by definition, lies over the totally nonnegative part of the Grassmannian. (This action may have period either or up on matrices, but it always has period down on the Grassmannian.) Postnikov’s motivation was to describe those matroids whose GGMS strata intersect this totally nonnegative part; it turns out that they are exactly the positroids, and the totally nonnegative part of each open positroid stratum is homeomorphic to a ball.
Now that we have defined the totally nonnegative part of the Grassmannian, we can explain the antecedents to Theorem 5.9. Postnikov ([Pos]) defined the totally nonnegative part of the Grassmannian as we have done above, by nonnegativity of all minors. Lusztig ([Lus98]) gave a different definition which applied to any . That the two notions agree is not obvious, and was established in [Rie09]. In particular, the cyclic symmetry seems to be special to Grassmannians.666Milen Yakimov has proven the stronger result that the standard Poisson structure on , from which the positroid stratification can be derived, is itself cyclic-invariant [Ya10].
Lusztig, using his definition, gave a stratification of by the projections of Richardson varieties. Theorem 3.8 of [Pos] (which relies on the results of [MarRie04] and [RieWi08]) states that Postnikov’s and Lusztig’s stratifications of coincide. This result says nothing about how the stratifications behave away from the totally nonnegative region. Theorem 5.9 can be thought of as a complex analogue of [Pos, Theorem 3.8]; it implies but does not follow from [Pos, Theorem 3.8].
We thank Konni Rietsch for helping us to understand the connections between these results.
1.3. Affine permutations, and the associated cohomology class of a positroid variety
Given a subvariety of a Grassmannian, one can canonically associate a symmetric polynomial in variables, in a couple of equivalent ways:
Sum, over partitions with , the Schur polynomial weighted by the number of points of intersection of with a generic translate of (the Schubert variety associated to the complementary partition inside the rectangle).
Take the preimage of in the Stiefel manifold of matrices of rank , and the closure inside matrices. (In the case this is the affine cone over a projective variety, and it seems worth it giving the name “Stiefel cone” in general.) This has a well-defined class in the equivariant Chow ring , which is naturally the ring of symmetric polynomials in variables.
The most basic case of is a Schubert variety , in which case these recipes give the Schur polynomial . More generally, the first construction shows that the symmetric polynomial must be “Schur-positive”, meaning a positive sum of Schur polynomials.
In reverse, one has ring homomorphisms
and one can ask for a symmetric function whose image is the class .
Theorem (Theorem 7.1777 Snider [Sni10] has given a direct geometric explanation of this result by identifying affine patches on with opposite Bruhat cells in the affine flag manifold, in a way that takes the positroid stratification to the Bruhat decomposition. Also, an analogue of this result for projected Richardson varieties in an arbitrary is established by He and Lam [HeLa]: the connection with symmetric functions is absent, but the cohomology classes of projected Richardson varieties and affine Schubert varieties are compared via the affine Grassmannian. ).
The cohomology class associated to a positroid variety can be represented by the affine Stanley function of its affine permutation, as defined in [Lam06].
This is a surprising result in that affine Stanley functions are not Schur-positive in general, even for this restricted class of affine permutations. Once restricted to the variables , they are! In Theorem 7.12 we give a much stronger abstract positivity result, for positroid classes in -equivariant -theory.
Our proof of Theorem 7.1 is inductive. In future work, we hope to give a direct geometric proof of this and Theorem 3.16, by embedding the Grassmannian in a certain subquotient of the affine flag manifold, and realizing the positroid decomposition as the transverse pullback of the affine Bruhat decomposition.
1.4. Quantum cohomology and toric Schur functions
In [BucKresTam03], Buch, Kresch, and Tamvakis related quantum Schubert calculus on Grassmannians to ordinary Schubert calculus on -step partial flag manifolds. In [Pos05], Postnikov showed that the structure constants of the quantum cohomology of the Grassmannian were encoded in symmetric functions he called toric Schur polynomials. We connect these ideas to positroid varieties:
Theorem (Theorem 8.1).
Let be the union of all genus-zero stable curves of degree which intersect a fixed Schubert variety and opposite Schubert variety . Suppose there is a non-trivial quantum problem associated to and . Then is a positroid variety: as a projected Richardson variety it is obtained by a pull-push from the 2-step flag variety considered in [BucKresTam03]. Its cohomology class is given by the toric Schur polynomial of [Pos05].
The last statement of the theorem is consistent with the connection between affine Stanley symmetric functions and toric Schur functions (see [Lam06]).
Our primary debt is of course to Alex Postnikov, for getting us excited about positroids. We also thank Michel Brion, Leo Mihalcea, Su-Ho Oh, Konni Rietsch, Frank Sottile, Ben Webster, Lauren Williams, and Milen Yakimov for useful conversations.
2. Some combinatorial background
Unless otherwise specified, we shall assume that nonnegative integers and have been fixed, satisfying .
2.1. Conventions on partitions and permutations
For integers and , we write to denote the interval , and to denote the initial interval . If , we let be the unique integer satisfying . We write for the set of -element subsets of . Thus denotes the set of -element subsets of .
As is well known, there is a bijection between and the partitions of contained in a box. There are many classical objects, such as Schubert varieties, which can be indexed by either of these -element sets. We will favor the indexing set , and will only discuss the indexing by partitions when it becomes essential, in §7.
We let denote the permutations of the set . A permutation is written in one-line notation as . Permutations are multiplied from right to left so that if , then . Thus multiplication on the left acts on values, and multiplication on the right acts on positions. Let be a permutation. An inversion of is a pair such that and . The length of a permutation is the number of its inversions. A factorization is called length-additive if .
The longest element of is denoted . The permutation is denoted (for Coxeter element). As a Coxeter group, is generated by the simple transpositions .
For , we let denote the parabolic subgroup of permutations which send to and to . A permutation is called Grassmannian (resp. anti-Grassmannian) if it is minimal (resp. maximal) length in its coset ; the set of such permutations is denoted (resp. ).
If and , then denotes the set . Often, we just write for when no confusion will arise. The map is a bijection when restricted to .
2.2. Bruhat order and weak order
We define a partial order on as follows. For and , we write if for .
We shall denote the Bruhat order, also called the strong order, on by and . One has the following well known criterion for comparison in Bruhat order: if then if and only if for each . Covers in Bruhat order will be denoted by and . The map is a poset isomorphism.
The (left) weak order on is the transitive closure of the relations
The weak order and Bruhat order agree when restricted to .
2.3. -Bruhat order and the poset .
The -Bruhat order [BeSo98, LasSchü82] on is defined as follows. Let and be in . Then -covers , written , if and only if and . The -Bruhat order is the partial order on generated by taking the transitive closure of these cover relations (which remain cover relations). We let denote the interval of in -Bruhat order. It is shown in [BeSo98] that every interval in is a graded poset with rank . We have the following criterion for comparison in -Bruhat order.
Theorem 2.1 ([BeSo98, Theorem A]).
Let . Then if and only if
implies and .
If , , and , then .
Define an equivalence relation on the set of -Bruhat intervals, generated by the relations that if there is a so that we have length-additive factorizations and . If , we let denote the equivalence class containing . Let denote the equivalence classes of -Bruhat intervals.
We discuss this construction in greater generality in [KnLamSp, §2]. To obtain the current situation, specialize the results of that paper to . The results we describe here are all true in that greater generality.
If then and this common ratio is in . Also .
This is obvious for the defining equivalences and is easily seen to follow for a chain of equivalences. ∎
We will prove a converse of this statement below as Proposition 2.4. The reader may prefer this definition of .
Every equivalence class in has a unique representative of the form where is Grassmannian. If is a -Bruhat interval, and is equivalent to with Grassmannian, then we have length-additive factorizations and with .
See [KnLamSp, Lemma 2.4] for the existence of a representative of this form. If and are two such representatives, then is in and both and are Grassmannian, so . Then so and we see that the representative is unique.
Finally, let be the representative with Grassmannian, and let . Set with . Since is Grassmannian, we have . Then the equation from Proposition 2.2 shows that as well. So the products and are both length-additive, as desired. ∎
We can use this observation to prove a more computationally useful version of the equivalence relation:
Given two -Bruhat intervals and , we have if and only if and common ratio lies in .
The forward implication is Proposition 2.2. For the reverse implication, let and be as stated. Let and be the representatives with Grassmannian. Since is in , and both are Grassmannian, then . Since , we deduce that . So and we have the reverse implication. ∎
We also cite:
Theorem 2.5 ([BeSo98, Theorem 3.1.3]).
If and with , then the map induces an isomorphism of graded posets .
3. Affine permutations, juggling patterns and positroids
Fix integers . In this section, we will define several posets of objects and prove that the posets are all isomorphic. We begin by surveying the posets we will consider. The objects in these posets will index positroid varieties, and all of these indexing sets are useful. All the isomorphisms we define are compatible with each other. Detailed definitions, and the definitions of the isomorphisms, will be postponed until later in the section.
We have already met one of our posets, the poset from §2.3.
The next poset will be the poset of bounded affine permutations: these are bijections such that , and . After that will be the poset of bounded juggling patterns. The elements of this poset are -tuples such that , where the subtraction of means to subtract from each element and our indices are cyclic modulo . These two posets are closely related to the posets of decorated permutations and of Grassmann necklaces, considered in [Pos].
We next consider the poset of cyclic rank matrices. These are infinite periodic matrices which relate to bounded affine permutations in the same way that Fulton’s rank matrices relate to ordinary permutations. Finally, we will consider the poset of positroids. Introduced in [Pos], these are matroids which obey certain positivity conditions.
The following is a combination of all the results of this section:
The posets , , , the poset of cylic rank matrices of type and the poset of positroids of rank on are all isomorphic.
The isomorphism between and cyclic rank matrices is Corollary 3.12; the isomorphism between and is Corollary 3.13; the isomorphism between and is Theorem 3.16; the isomorphism between and positroids is Proposition 3.21.
3.1. Juggling states and functions
Define a (virtual) juggling state as a subset whose symmetric difference from is finite. (We will motivate this and other juggling terminology below.) Let its ball number be , where . Ball number is the unique function on juggling states such that for , the difference in ball numbers is , and has ball number zero.
Call a bijection a (virtual) juggling function if for some (or equivalently, any) , the set is a juggling state. It is sufficient (but not necessary) that be bounded. Let be the set of such functions: it is easy to see that is a group, and contains the element . Define the ball number of as the ball number of the juggling state , and denote it for reasons to be explained later.
is a group homomorphism.
We prove what will be a more general statement, that if is a juggling state with ball number , and a juggling function with ball number , then is a juggling state with ball number . Proof: if we add one element to , this adds one element to , and changes the ball numbers of by . We can use this operation and its inverse to reduce to the case that , at which point the statement is tautological.
Now let , and apply the just-proven statement to . ∎
For any bijection , let
and if , call it the juggling state of at time . By the homomorphism property just proven , which says that every state of has the same ball number (“ball number is conserved”). The following lemma lets one work with juggling states rather than juggling functions:
Say that a juggling state can follow a state if , and . In this case say that a -throw takes state to state .
Then a list is the list of states of a juggling function iff can follow for each . In this case the juggling function is unique.
If the arise from a juggling function , then the condition is satisfied where the element added to is . Conversely, one can construct as . ∎
In fact the finiteness conditions on juggling states and permutations were not necessary for the lemma just proven. We now specify a further finiteness condition, that will bring us closer to the true functions of interest.
The following two conditions on a bijection are equivalent:
there is a uniform bound on , or
there are only finitely many different visited by .
If they hold, is a juggling function.
Assume first that has only finitely many different states. By Lemma 3.3, we can reconstruct the value of from the states . So takes on only finitely many values, and hence is uniformly bounded.
For the reverse, assume that for all . Then , and . Since is bijective, we can complement the latter to learn that . So , and similarly each , is trapped between and . There are then only possibilities, all of which are juggling states. ∎
In the next section we will consider juggling functions which cycle periodically through a finite set of states.
Define the height of the juggling state as
a sort of weighted ball number. We can now motivate the notation , computing ball number as an average:
Let and let . Then
In particular, if satisfies the conditions of Lemma 3.4, then for any ,
This equality also holds without taking the limit, if .
It is enough to prove the first statement for , and add the many equations together. They are of the form
To see this, start with , and use to calculate . The three sets to consider are
By its definition, . And . The equation follows.
For the second, if only visits finitely many states then the difference in heights is bounded, and dividing by kills this term in the limit. ∎
We now motivate these definitions from a juggler’s point of view. The canonical reference is [Pol03], though our setting above is more general than considered there. All of these concepts originated in the juggling community in the years 1985-1990, though precise dates are difficult to determine.
Consider a idealized juggler who is juggling with one hand888or as is more often assumed, rigidly alternating hands, making one throw every second, of exactly one ball at a time, has been doing so since the beginning of time and will continue until its end. If our juggler is only human (other than being immortal) then there will be a limit on how high the throws may go.
Assume at first that the hand is never found empty when a throw is to be made. The history of the juggler can then be recorded by a function
The number is usually called the throw at time . If ever the juggler does find the hand empty i.e. all the balls in the air, then of course the juggler must wait one second for the balls to come down. This is easily incorporated by taking , a -throw.
While these assumptions imply that is a juggling function, they would also seem to force the conclusion that , i.e. that balls land after they are thrown. Assuming that for a moment, it is easy to compute the number of balls being juggled in the permutation : at any time , count how many balls were thrown at times that are still in the air, . This is of course our formula for the ball number, in this special case. The formula then says that balls are neither created nor destroyed.
The state of at time is the set of times in the future (of ) that balls in the air are scheduled to land. (This was introduced to study juggling by the first author and, independently, by Jack Boyce, in 1988.) The “height” of a state does not seem to have been considered before.
Thus, the sub-semigroup of where encodes possible juggling patterns. Since we would like to consider as a group (an approach pioneered in [EhRe96]), we must permit . While it may seem fanciful to view this as describing juggling with antimatter, the “Dirac sea” interpretation of antimatter is suggestive of the connection with the affine Grassmannian.
3.2. Affine permutations
Let denote the group of bijections, called affine permutations, satisfying
Plainly this is a subgroup of . This group fits into an exact sequence
where for , we define the translation element by for . The map is evident. We can give a splitting map by extending a permutation periodically. By this splitting, we have , so every can be uniquely factorized as with and .
An affine permutation is written in one-line notation as (or occasionally just as ). As explained in Section 1.2, jugglers instead list one period of the periodic function (without commas, because very few people can make -throws999The few that do sometimes use to denote throws , which prompts the question of what words are jugglable. Michael Kleber informs us that THEOREM and TEAKETTLE give valid juggling patterns. and higher), and call this the siteswap. We adopt the same conventions when multiplying affine permutations as for usual permutations. The ball number is always an integer; indeed . Define
so that is the Coxeter group with simple generators , usually called the affine symmetric group.101010One reason the subgroup is more commonly studied than is that it is the Coxeter group ; its relevance for us, is that it indexes the Bruhat cells on the affine flag manifold for the group . In §7 we will be concerned with the affine flag manifold for the group , whose Bruhat cells are indexed by all of . Note that if and then the product is in . There is a canonical bijection between the cosets and . The group has a Bruhat order “” because it is a Coxeter group . This induces a partial order on each , also denoted .
An inversion of is a pair such that and . Two inversions and are equivalent if and for some integer . The number of equivalence classes of inversions is the length of . This is sort of an “excitation number” of the juggling pattern; this concept does not seem to have been studied in the juggling community (though see [EhRe96]).
An affine permutation is bounded if for . We denote the set of bounded affine permutations by . The restriction of the Bruhat order to is again denoted .
The subset is a lower order ideal in . In particular, is graded by the rank function .
Suppose and . Then is obtained from by swapping the values of and for each , where and . By the assumption on the boundedness of , we have and . Thus . ∎
Postnikov, in [Pos, §13], introduces “decorated permutations”. A decorated permutation is an element of , with each fixed point colored either or . There is an obvious bijection between the set of decorated permutations and : Given an element , form the corresponding decorated permutation by reducing modulo and coloring the fixed points of this reduction or according to whether or respectively. In [Pos, §17], Postnikov introduces the cyclic Bruhat order, , on those decorated permutations corresponding to elements of . From the list of cover relations in [Pos, Theorem 17.8], it is easy to see that is anti-isomorphic to .
In the case there are already bounded affine permutations, but only up to cyclic rotation. In Figure 1 we show the posets of siteswaps, affine permuations, and decorated permutations, each modulo rotation. Note that the cyclic symmetry is most visible on the siteswaps, and indeed jugglers draw little distinction between cyclic rotations of the “same” siteswap.
3.3. Sequences of juggling states
A -sequence of juggling states is a sequence such that for each , we have that follows , where the indices are taken modulo . Let denote the set of such sequences.
Let . Then the sequence of juggling states
is periodic with period . Furthermore for each , (a) , and (b) . Thus
The map is a bijection between and .
We now discuss another way of viewing -sequences of juggling states which will be useful in §5.1. Let be a -ball virtual juggling state. For every integer , define
These satisfy the following properties:
is either or , according to whether or respectively,
for sufficently positive, and
for sufficiently negative.
Conversely, from such a sequence one can construct a -ball juggling state.
Let and be two -ball juggling states. Define a matrix by .
The state can follow if and only if or for all and there is no submatrix for which .
It is easy to check that if and only if for and for . If this holds, it immediately follows that or for all and that there is no for which while .
Conversely, suppose that or for all and there is no for which . Then we claim that there is no for which and . Proof: suppose there were. If , then we are done by our hypothesis; if then , contradicting that or . Since or , we have a contradiction either way. This establishes the claim. Now, we know that for sufficently positive and for sufficently negative, so there must be some such that for and for . Then can follow . ∎
It is immediate to extend this result to a sequence of juggling states. Let be the group of juggling functions introduced in §3.1 and let be those juggling functions with ball number . For any in , let be the corresponding -sequence of juggling states. Define an matrix by . Then, applying Lemma 3.9 to each pair of rows of gives:
The above construction gives a bijection between and matrices such that
for each , there is an such that for all ,
for each , there is an such that for all ,