aa r X i v : . [ m a t h . C O ] M a r A DIRECTED GRAPH STRUCTURE OF ALTERNATING SIGNMATRICES
MASATO KOBAYASHI
Abstract.
We introduce a new directed graph structure into the set of alter-nating sign matrices. This includes Bruhat graph (Bruhat order) of the sym-metric groups as a subgraph (subposet).Drake-Gerrish-Skandera (2004, 2006) gave characterizations of Bruhat orderin terms of total nonnegativity (TNN) and subtraction-free Laurent (SFL) ex-pressions for permutation monomials. With our directed graph, we extend theiridea in two ways: first, from permutations to alternating sign matrices; second, q -analogs (which we name q TNN and q SFL properties). As a by-product, weobtain a new kind of permutation statistic, the signed bigrassmannian statistics,using Dodgson’s condensation on determinants.
Date : March 21, 2019.2000
Mathematics Subject Classification.
Primary:15B36; Secondary:05A05, 05B20, 11C20.
Key words and phrases.
Alternating sign matrices, Bigrassmannian permutations, Bruhatorder, determinant, Essential sets, Permutation Statistics, Subtraction-Free Laurent expressions,Total nonnegativity.This was already published as M. Kobayashi, A directed graph structure of alternating signmatrices, Linear Algebra and its Applications (2017), 164-190.
Contents
1. Introduction 21.1. Bruhat order 21.2. Main results 31.3. Outline 31.4. Additional note 32. Alternating sign matrices 42.1. Alternating sign matrices 42.2. Corner sum matrices 53. Bruhat graph and ASM graph 73.1. Bruhat graph 73.2. ASM order 83.3. Essential rectangles 83.4. Key Lemma 113.5. Essential points 154. Total nonnegativity and (SFL) property 164.1. Total nonnegativity 164.2. (SFL) property 184.3. Drake-Gerrish-Skandera’s characterizations 185. Main results 185.1. Bigrassmannian statistic 185.2. ( q TNN) and ( q SFL) properties 195.3. Characterizations of ASM order 205.4. Corollaries 215.5. Signed bigrassmannian statistics 236. Concluding remarks 26References 261.
Introduction
Bruhat order.
Bruhat order has been of great importance in the combina-torial matrix theory; there are many equivalent characterizations of this order. Forexample, one is the transitive closure of the binary relation u → v on S n to mean v = ut for some transposition t and ℓ ( u ) < ℓ ( v ) (with ℓ the number of inversions).Other variations are: • Entrywise order on Corner sum matrices; for example, see Brualdi-Deaett[3] and Fortin [8]. • Lascoux-Sch¨utzenberger’s monotone triangles [13].In addition to this list, Drake-Gerrish-Skandera [6, 7] found several new charac-terizations of Bruhat order in terms of permutation monomials:
DIRECTED GRAPH STRUCTURE OF ASMS 3
Figure 1.
ASMs and Related ideasTotal nonnegativityCorner summatrix Alternating sign matrices(Bruhat order, ASM graph) determinantsbigrassmannianpermutations
Fact 1.1.
Let u, v ∈ S n . Then the following are equivalent:(1) u ≤ v in Bruhat order.(2) the polynomial x u (1) · · · x nu ( n ) − x v (1) · · · x nv ( n ) is TNN.(3) the polynomial x u (1) · · · x nu ( n ) − x v (1) · · · x nv ( n ) has (SFL) property.Here TNN and SFL abbreviate “Totally NonNegative” and “Subtraction-FreeLaurent expression”, respectively; we give details of these terms later.1.2. Main results.
The aim of this article is simply to generalize Drake-Gerrish-Skandera’s result above in two ways (Theorem 5.9); first, permutations to alter-nating sign matrices (ASMs); second, we will establish a q -analog of their result.We also observe some byproducts on permutation statistics (Theorems 5.1 and5.14). For this purpose, we introduce a new directed graph structure to ASMs asin the title; we call it ASM graph (Figure 1).1.3.
Outline.
This articles consists of six sections. Section 2 serves preliminarieson permutations and alternating sign matrices. Section 3 gives a precise definitionof ASM graph with notions of essential rectangles and bigrassmannian statistics;in particular, Key Lemma 3.18 will play a role in the sequel. In Section 4, wereview Total nonnegativity and Subtraction-Free Laurent property. In Section 5,we give proofs of main results. We end with the conclusion remark in Section 6.To better understand the global picture of our discussion, it is helpful to keepFigure 2 in mind.1.4.
Additional note.
At the time of writing this article, the author found thatthere are overlap with the recent articleR. Brualdi, M. Schroeder, Alternating sign matrices and theirBruhat order, to appear in Discrete Math.Brualdi and Schroeder discuss the sequential construction of an ASM from the unitmatrix (corresponding to our directed graph structure) as well as an enumerative
MASATO KOBAYASHI
Figure 2.
Global picture of our discussionPermutation matrices S n Bruhat orderBruhat graphgrading:inversion number O O order isomorphic (cid:15) (cid:15) ⊆ Alternating sign matrices A n ASM (Bruhat) orderASM graphgrading:bigrassmannian statistics O O order isomorphic (cid:15) (cid:15) Corner sum matrices f S n entrywise order ⊆ Corner sum matrices f A n entrywise orderproperty of B-rank function for ASMs (corresponding to bigrassmannian statisticsin our terminology). acknowledgment. The author would like to thank the editor as well as the anonymous referee forhelpful comments for improvement of the manuscript.2.
Alternating sign matrices
For a positive integer n , let [ n ] denote the set { , , . . . , n } . Throughout thisarticle, we assume that n ≥ S n we mean the sym-metric group on [ n ]. To represent permutations, we often use one-line notation :“ u = i · · · i n ” with i k ∈ [ n ] means u ( k ) = i k . For instance, u = 231 means u (1) = 2 , u (2) = 3 and u (3) = 1. Below, A = ( a ij ) and B = ( b ij ) are squarematrices of size n unless otherwise specified. For convenience, we write a ij as wellas A ( i, j ) for a matrix entry of A .2.1. Alternating sign matrices.
We begin with definitions of permutation ma-trices and alternating sign matrices.
Definition 2.1.
We say that A is a permutation matrix (PM) if there exists aunique permutation u ∈ S n such that a ij = 1 if j = u ( i ) and a ij = 0 otherwise.In this way, we often identify a permutation and a permutation matrix. DIRECTED GRAPH STRUCTURE OF ASMS 5
Figure 3. ( A , ≤ ) ✈✈✈✈✈ ❍❍❍❍❍ ❍❍❍❍❍ ✈✈✈✈✈ − ✈✈✈✈✈ ❍❍❍❍❍ ❍❍❍❍❍ ✈✈✈✈✈ Definition 2.2.
We say that A is an alternating sign matrix (ASM) if for all( i, j ) ∈ [ n ] , we have a ij ∈ {− , , } , j X k =1 a ik ∈ { , } , i X k =1 a kj ∈ { , } and n X k =1 a ik = n X k =1 a kj = 1 . Denote by A n the set of all alternating sign matrices of size n .Note that every PM is an ASM. Say an ASM is proper if it is not a PM; in otherwords, an ASM is proper if and only if it has a − A ; the only one matrix in the middle is proper.2.2. Corner sum matrices.
MASATO KOBAYASHI
Figure 4. ( e A , ≤ ) ✇✇✇✇✇ ●●●●● ●●●●● ✇✇✇✇✇ ✇✇✇✇✇ ●●●●● ●●●●● ✇✇✇✇✇ Definition 2.3.
The corner sum matrix of A ∈ A n is the n by n matrix e A definedby e A ( i, j ) = X p ≤ i,q ≤ j a pq for all i, j . Denote by e A n the set of all such matrices. Example 2.4.
For A = − , we have e A = . Remark 2.5. (1) Entries of each corner sum matrix are weakly increasing along rows andcolumns: e A ( i, j ) ≤ e A ( k, l ) if i ≤ k and j ≤ l . DIRECTED GRAPH STRUCTURE OF ASMS 7 (2) It is convenient to define a ij = 0 and e A ( i, j ) = 0 whenever i or j is 0.Then, we can recover each entry a ij from entries of e A : a ij = e A ( i, j ) + e A ( i − , j − − e A ( i, j − − e A ( i − , j ) for i, j ≥ . The correspondence A ↔ e A between A n and e A n is in fact a bijection; seeFigures 3 and 4, for example.The following criterion will be useful later. Fact 2.6 (Robbins-Rumsey [15, p.172, Lemma 1]) . Let X be a square matrixof size n . Then X ∈ e A n if and only if X ( i, n ) = X ( n, i ) = i for all i and X ( i, j ) − X ( i − , j ) ∈ { , } , X ( i, j ) − X ( i, j − ∈ { , } for all i, j .3. Bruhat graph and ASM graph
In this section, we give a precise definition of
ASM graph ; this is a directed graphstructure of ASMs as in the title of this article. We first review the definition ofBruhat graph on permutations; we will see that it is a certain subgraph of ASMgraph.3.1.
Bruhat graph.
For natural numbers i < j ≤ n , let t ij denote the transpo-sition interchanging i and j . Say a pair ( i, j ) is an inversion of a permutation u ∈ S n if i < j and u ( i ) > u ( j ). Let ℓ ( u ) be the number of inversions of u . Write u → v if v = ut ij and ℓ ( u ) < ℓ ( v ) (equivalently, ( i, j ) is an inversion of v ). Thedirected graph ( S n , → ) is the Bruhat graph . Example 3.1.
We have the edge relation 1342 → → interchanging first and fourth columns (first and third rows). Definition 3.2.
Define
Bruhat order u ≤ v in S n if there exists a directed pathfrom u to v .This is indeed a partial order on S n . Here are more details: Fact 3.3 (Chain Property) . ( S n , ≤ ) is a graded poset ranked by ℓ . In other words,if u ≤ v , then there exists a directed path u = u → u → u → · · · → u k = v such that ℓ ( u i ) − ℓ ( u i − ) = 1.We wish to extend Bruhat order to ASMs (recall that every PM is an ASM).However, we have to take care of the following two points: MASATO KOBAYASHI • Transposing columns or rows of an ASM does not necessarily produce anASM. Thus, we need to modify a definition of the edge relation. • Find a rank function on ASMs, instead of the inversion number, such thatit is monotonically increasing along those directed edges.We solve these problems with a new definition of a directed edge relation using corner sum matrices and bigrassmannian statistics .3.2.
ASM order.
Make sure that there is an equivalent characterization of Bruhatorder in terms of corner sum matrices (rather than entries of PMs):
Fact 3.4.
The following are equivalent:(1) u ≤ v in Bruhat order in S n .(2) e u ( i, j ) ≥ e v ( i, j ) for all i, j ∈ [ n ].This idea naturally extends to ASMs: Definition 3.5.
Define
ASM order A ≤ B in A n if e A ( i, j ) ≥ e B ( i, j ) for all i, j ∈ [ n ].By abuse of language, we also call this “Bruhat order”. Hence ( A n , ≤ ) is now aposet. Remark 3.6.
Indeed, ( A n , ≤ ) is a finite distributive lattice as the MacNeille com-pletion of Bruhat order (the smallest lattice which contains ( S n , ≤ ) as a subposet).See Reading [14] for some more details.3.3. Essential rectangles.
As before, let A be an ASM. Consider integers i, j, k, l ∈ [ n ] such that i < j and k < l . Let R klij = { ( p, q ) ∈ [ n ] | i ≤ p < j and k ≤ q < l } be rectangular positions in a matrix (here, i ≤ p and k ≤ q are weak inequalitieswhile p < j and q < l are strict). Definition 3.7.
We say that R klin is an essential rectangle for A if e A ( p, k ) = e A ( p, k − , e A ( p, l ) = e A ( p, l −
1) + 1 , e A ( i, q ) = e A ( i − , q ) , and e A ( j, q ) = e A ( j − , q ) + 1for all ( p, q ) ∈ R klij . Similarly, say R klij is a dual essential rectangle for A if e A ( p, k ) = e A ( p, k −
1) + 1 , e A ( p, l ) = e A ( p, l − , e A ( i, q ) = e A ( i − , q ) + 1 , and e A ( j, q ) = e A ( j − , q )for all ( p, q ) ∈ R klij . We call such conditions (dual) essential conditions . Denote by E ( A ) ( E ∗ ( A )) the set of such (dual) rectangles for A . DIRECTED GRAPH STRUCTURE OF ASMS 9
Recall that adjacent entries of any corner sum matrix differs only by 0 or 1.These conditions above describe “boundary conditions” on these rectangular po-sitions. Note: we understand e A ( p, q ) = 0 if p or q is 0; we often omit these zeroentries when we write a corner sum matrix. Example 3.8.
On the one hand, the permutation 4312 has an essential rectangle R since g . On the other hand, the permutation 1342 has a dual essential rectangle R since g . As we see, underlined positions indicate such rectangles.
Proposition 3.9.
Let u ∈ S n and i < j . Then the following are equivalent: (1) ( i, j ) is an inversion of u . (2) R u ( j ) ,u ( i ) i,j is an essential rectangle for u .Proof. If ( i, j ) is an inversion of u , then there exist two 1s at ( i, u ( i )) and ( j, u ( j ))positions in the permutation matrix u . It follows from the definition of a corner summatrix that R u ( j ) u ( i ) ij satisfies the essential conditions described above. Conversely,if R u ( j ) u ( i ) ij is an essential rectangle for u , then it is necessarily that u ( j ) < u ( i ). (cid:3) Definition 3.10.
For i < j and k < l , let e R klij be the n by n matrix such thatits ( p, q )-entry is 1 if ( p, q ) ∈ R klij or 0 otherwise. Define a rectangular operator e r klij : e A n → e A n e r klij ( e A ) = e A + e R klij if R klij ∈ E ( A ) , e A − e R klij if R klij ∈ E ∗ ( A ) ,A otherwise.So this operator changes entries of a consecutive submatrix of entries of a cornersum matrix. Example 3.11. g e r ( g | {z } g + | {z } e R . Similarly, define an operator r klij : A n → A n with r klij ( A ) being the ASM whosecorner sum matrix is e r klij ( e A ). Remark 3.12. (1) Let us be careful: whenever R klij ∈ E ( A ), is the resulting matrix e A + e R klij an element of e A n ? Yes. Indeed, adjacent entries of e A + e R klij differ only by0 or 1 (sharing the n -th row and column entries of e A ). Fact 2.6 guaranteesthat e A + e R klij is a corner sum matrix for some (unique) ASM.(2) Observe that r klij is an involution, i.e., ( r klij ) A = A .With this idea, it is natural to introduce the following statistic for ASMs as (thenegative of) a sum of entries of corner sum matrices. Definition 3.13.
For i, j , let i ∧ j = min { i, j } . For A ∈ A n , define the bigrass-mannian statistic β ( A ) = n X i,j =1 ( i ∧ j ) − n X i,j =1 e A ( i, j ) . Here the constant P i ∧ j comes for normalization so that β ( e ) = 0 where e isthe unit of S n so that e e ( i, j ) = i ∧ j .Observe the following dichotomy: for each R klij ∈ E ( A ) ∪ E ∗ ( A ), we have either β ( r klij A ) < β ( A ) ⇐⇒ R klij ∈ E ( A ) or β ( r klij A ) > β ( A ) ⇐⇒ R klij ∈ E ∗ ( A ). Withnotions of essential rectangles and this statistic, we are now ready to introduceASM graph as a generalization of Bruhat graph. Definition 3.14.
Define an edge relation A kl → ij B in A n if B = r klij ( A ) and β ( A ) <β ( B ). By A → B we mean A kl → ij B for some i, j, k, l . Call the directed graph( A n , → ) ASM graph .It naturally induces the same directed graph structure on e A n ; by abuse of lan-guage, we call it ASM graph as well.As shown above, every edge in Bruhat graph is also an edge in ASM graph; seeFigure 5. In terms of this new graph, we may characterize ASM order as follows: Proposition 3.15.
The following are equivalent:
DIRECTED GRAPH STRUCTURE OF ASMS 11
Figure 5. ( A , → ) : : ✉✉✉✉✉✉ d d ■■■■■■ X X d d ■■■■■■ O O n n : : ✉✉✉✉✉✉ O O − : : ✉✉✉✉✉✉ d d ■■■■■■ d d ■■■■■■ : : ✉✉✉✉✉✉ (1) A ≤ B in ASM order. (2) There exists a directed path from A to B . Key Lemma.
We defined the edge relation for two ASMs in terms of theircorner sum matrices. Along this relation, what happens back to entries of the twoASMs? Key Lemma 3.18 below answers this question completely; it will play akey role to prove main results in Section 5. Before that, we take auxiliary twosteps with the following lemmas.
Lemma 3.16 (nonpositivity) . Let B ∈ A n . Suppose R klij ∈ E ( B ) is given. Then, b ik ≤ and b jl ≤ .Proof. Suppose R klij ∈ E ( B ). Thanks to one of the essential conditions e B ( i, k ) = e B ( i, k − b ik = e B ( i, k )+ e B ( i − , k − − e B ( i, k − − e B ( i − , k ) = e B ( i − , k − − e B ( i − , k ) ≤ . Figure 6. ( f A , → ) : : ✈✈✈✈✈ d d ❍❍❍❍❍ X X d d ❍❍❍❍❍ O O n n : : ✈✈✈✈✈ O O : : ✈✈✈✈✈ d d ❍❍❍❍❍ d d ❍❍❍❍❍ : : ✈✈✈✈✈ Moreover, two of essential conditions e B ( j, l −
1) = e B ( j − , l − e B ( j − , l ) = e B ( j − , l −
1) + 1 imply that b jl = e B ( j, l ) + e B ( j − , l − − e B ( j, l − − e B ( j − , l )= e B ( j, l ) − e B ( j − , l − − ≤ . (cid:3) Lemma 3.17 (nonnegativity) . Let B ∈ A n . Suppose R klij ∈ E ( B ) is given. Then, b il ≥ and b jk ≥ .Proof. Thanks to one of essential conditions e B ( i, l ) = e B ( i, l −
1) + 1, we have b il = e B ( i, l )+ e B ( i − , l − − e B ( i, l − − e B ( i − , l ) = e B ( i − , l − − e B ( i − , l )+1 ≥ . It is similar to show that b jk ≥ (cid:3) These two lemmas assert that each of b ik , b il , b jk , b jl can take two values. Intotal, there are 16 cases as listed in Table 1. DIRECTED GRAPH STRUCTURE OF ASMS 13
Table 1.
16 kinds of edge relations A → B in ASM graphtype b ik b il b jk b jl a ik a il a jk a jl type b ik b il b jk b jl a ik a il a jk a jl − −
10 1 − −
10 1 − − − − − − − − − − − − −
10 0 − − −
10 0 − − − − − − − − − − − − Key Lemma 3.18.
Let B ∈ A n and R klij ∈ E ( B ) . Consider a square matrix A ofsize n . Then, the following are equivalent: (1) A kl → ij B . (2) The entries ( a ik , a il , a jk , a jl ) satisfy (cid:18) a ik a il a jk a jl (cid:19) − (cid:18) b ik b il b jk b jl (cid:19) = (cid:18) − − (cid:19) as listed in Table 1. Moreover, if ( p, q )
6∈ { ( i, k ) , ( i, l ) , ( j, k ) , ( j, l ) } , then a pq = b pq .Proof. (1) = ⇒ (2): Suppose e A = e B + e R klij so that e A ( p, q ) = e B ( p, q ) if and only if( p, q ) R klij . Thus, equalities a ik = e A ( i, k ) + e A ( i − , k − − e A ( i − , k ) − e A ( i, k −
1) and b ik = e B ( i, k ) + e B ( i − , k − − e B ( i − , k ) − e B ( i, k − a ik − b ik = e A ( i, k ) − e B ( i, k ) = 1 (the other six terms are gone). Similarly, a il = e A ( i, l ) + e A ( i − , l − − e A ( i − , l ) − e A ( i, l −
1) and b il = e B ( i, l ) + e B ( i − , l − − e B ( i − , l ) − e B ( i, l − a il − b il = − e A ( i, l − e B ( i, l −
1) = −
1. In the same way, a jk − b jk = − a jl = e A ( j, l ) + e A ( j − , l − − e A ( j − , l ) − e A ( j, l −
1) and b jl = e B ( j, l ) + e B ( j − , l − − e B ( j − , l ) − e B ( j, l − a jl − b jl = e A ( j − , l − − e B ( j − , l −
1) = 1. For other ( p, q ), observethat |{ ( p, q ) , ( p − , q − , ( p − , q ) , ( p, q − } ∩ R klij | is either 0, 2 or 4. If it is 0or 4, then clearly a pq = b pq follows. If it is 2, then either p ∈ { i, j } or q ∈ { k, l } .Here suppose p = i and q
6∈ { k, l } so that a pq − b pq = e A ( p, q ) − e B ( p, q ) − ( e A ( p, q − − e B ( p, q − − . It is analogous to verify other cases.(2) = ⇒ (1): We can reverse most of the proof above. (cid:3) Table 1 indicates such 16 edge relations; note that only the type 1 occurs inBruhat graphs. It is convenient to say that a 2 by 2 minor in an ASM is inter-changeable if it is one of the 32 patterns in the table.
Example 3.19.
Let B = − − be an ASM of size 5. Its cornersum matrix is . Here the underlined part refers to R . Then, DIRECTED GRAPH STRUCTURE OF ASMS 15 we have A = − → − − = B. This is type 9.3.5.
Essential points.
As seen in the previous example, an essential rectanglecan be of size 1.
Definition 3.20.
We say that an essential rectangle R klij is an essential point if j = i + 1 and l = k + 1 (so that | R klij | = 1). Remark 3.21.
Here, we have a specific reason to coin the term “essential point”;Fulton [11] defined essential sets for permutations as follows:Ess( w ) = { ( i, j ) ∈ [ n − | i < w − ( j ) , j < w ( i ) , w ( i + 1) ≤ j, w − ( j + 1) ≤ i } . We may rephrase these four conditions in terms of corner sum matrices: For each( i, j ) ∈ [ n − , the following equivalences hold (as easily checked):(1) i < w − ( j ) ⇐⇒ e w ( i − , j ) = e w ( i, j ) . (2) j < w ( i ) ⇐⇒ e w ( i, j −
1) = e w ( i, j ) . (3) w ( i + 1) ≤ j ⇐⇒ e w ( i + 1 , j ) = e w ( i, j ) + 1 . (4) w − ( j + 1) ≤ i ⇐⇒ e w ( i, j + 1) = e w ( i, j ) + 1 . Thus, ( i, j ) is an essential point of w if and only if ( i, j ) is an element of Ess( w ).As a consequence of Key Lemma, there is a one-to-one correspondence betweenessential points of B and ASMs covered by B . Hence every covering relation inASM order is an edge relation of ASM graph.Define a permutation w to be bigrassmannian if there exists a unique pair ( i, j ) ∈ [ n − with w − ( i ) > w − ( i + 1) and w ( j ) > w ( j + 1). Proposition 3.22.
For A ∈ A n , the following are equivalent: (1) A is a bigrassmannian permutation. (2) A has exactly one essential point.Proof. (Sketch) Both are equivalent to what we call join-irreducibility ; see Lascoux-Sch¨utzenberger [13] for details of equivalence of bigrassmannian and join-irreducibility.Recall from the theory of finite distributive lattices [14] that an element is join-irreducible in such a lattice if and only if it covers exactly one element. (cid:3) For example, g has exactly one essential point so that1342 is bigrassmannian. Proposition 3.23. (1) (Chain Property) If A ≤ B , then there exists a directed path A → A → A → · · · → A k = B such that β ( A i ) − β ( A i − ) = 1 for all i . (2) For each A ∈ A n , we have β ( A ) = |{ B ∈ A n | B ≤ A and B is bigrassmannian }| . Proof. (Sketch) As Reading reviewed [14], ( A n , ≤ ) is (isomorphic to) a finite dis-tributive lattice graded by |{ B ∈ A n | B ≤ A and B is join-irreducible }| . Since β ( e ) = 0 ( e the minimum element) and β increases by one along every coveringrelation, this function must coincide with β . As a result, these two assertionsfollow. (cid:3) For this reason, we call β bigrassmannian statistics . We will show more explicitformulas for β in Section 5.4. Total nonnegativity and (SFL) property
Toward our main result, we now need key ideas: total nonnegativity and subtraction-free Laurent (SFL) property. Although these are classical topics in applications ofLinear Algebra (as Ando [1]), here let us review precise definitions of such ideas.4.1.
Total nonnegativity.
Let A be a real n by n matrix. Definition 4.1.
We say that A = ( a ij ) is totally nonnegative (TNN) if the deter-minant for every square submatrix of A is nonnegative. Remark 4.2.
Some authors use the term “totally positive” to mean the samething. Here we followed Drake-Gerrish-Skandera [6, 7].Let x , . . . , x nn be commutative variables and f ( x , · · · , x nn ) a real polyno-mial. When no confusion arises, we simply write f ( x ) to mean the polynomial f ( x , . . . , x nn ). Similarly, for a real matrix A = ( a ij ), we write f ( A ) to mean thereal number f ( a , . . . , a nn ). Definition 4.3.
We say that a polynomial f ( x ) is totally nonnegative (TNN) ifwhenever A is a TNN matrix of size n , then f ( A ) ≥ Remark 4.4.
In particular, if this is the case, then we have a ij ≥ i, j )because a ij is itself the determinant of a 1 by 1 submatrix. Definition 4.5.
Given u ∈ S n , let x u denote the monomial x u (1) · · · x nu ( n ) . Wecall it the permutation monomial for u . DIRECTED GRAPH STRUCTURE OF ASMS 17
Example 4.6.
Let u = and v = . Then x u − x v = x x x − x x x is TNN since we have the inequality a a a − a a a = a (cid:12)(cid:12)(cid:12)(cid:12) a a a a (cid:12)(cid:12)(cid:12)(cid:12) ≥ A = ( a ij ).Now we extend total nonnegativity for ASMs. As above, let x , . . . , x nn becommutative variables. For our purpose, consider a rational function g ( x ) = g ( x , · · · , x nn ) rather than a polynomial. Definition 4.7.
We say that a rational function g ( x ) is totally nonnegative (TNN)if whenever A is a TNN matrix of size n and moreover g ( A ) is defined, then g ( A ) ≥ g ( x ) is indeed a polynomial, then this definition coincides with the totalnonnegativity above. Definition 4.8.
For each A ∈ A n , introduce the ASM (Laurent) monomial x A := n Y i,j =1 x a ij ij . Apparently, this idea includes permutation monomials.
Example 4.9.
Let B = and C = − .Then g ( x ) = x B − x C = x x x − x x x − x x is TNN since we have theinequality g ( A ) = a a a − a a a − a a = a a (cid:12)(cid:12)(cid:12)(cid:12) a a a a (cid:12)(cid:12)(cid:12)(cid:12) a ≥ A = ( a ij ) such that a = 0.This example suggests the following consequence of Key Lemma. If A → B ,then there exists a unique ( i, j, k, l ) ∈ [ n ] such that { ( p, q ) ∈ [ n ] | a pq = b pq } = { ( i, k ) , ( i, l ) , ( j, k ) , ( j, l ) } . It leads to a decomposition of a difference of ASM monomials: Set x AB := Y a pq = b pq x a pq pq and x E ( A,B ) := Y a pq = b pq x a pq pq − Y a pq = b pq x b pq pq . Clearly, the latter corresponds to interchangeable entries of A and B . These tworational functions give the decomposition x A − x B = x AB x E ( A,B ) . Observe that,in any case, x E ( A,B ) is a product of (cid:12)(cid:12)(cid:12)(cid:12) x ik x il x jk x jl (cid:12)(cid:12)(cid:12)(cid:12) and a Laurent monomial in thesefour variables as x E ( A,B ) = x a ik ik x a il il x a jk jk x a jl jl − x b ik ik x b il il x b jk jk x b jl jl = x a ik ik x a il il x a jk jk x a jl jl − x a ik − ik x a il +1 il x a jk +1 jk x a jl − jl = x a ik ik x a il il x a jk jk x a jl jl (cid:12)(cid:12)(cid:12)(cid:12) x ik x il x jk x jl (cid:12)(cid:12)(cid:12)(cid:12) x ik x jl . (SFL) property. Let f ( x ) be a real polynomial. Definition 4.10.
We say that f ( x ) has Subtraction-Free Rational (SFR) propertyif f ( x ) has a rational expression in minors of the matrix x = ( x ij ) ni,j =1 such that itsdenominator and numerator do not contain any subtraction. Also say that f ( x )has Subtraction-Free Laurent (SFL) property if f ( x ) has (SFR) property with arational expression such that its denominator is a monomial in minors of x .We could define these properties for rational functions of x , . . . , x nn in theexactly same way. For example, g ( x ) = x x x − x x x − x x has (SFR)and (SFL) properties as mentioned above.4.3. Drake-Gerrish-Skandera’s characterizations.
In the last two subsections,we reviewed two properties on polynomials. What is the relation between (TNN),(SFL) properties and Bruhat order? Drake-Gerrish-Skandera [6, 7] established thefollowing equivalence:
Fact 4.11.
Let u, v ∈ S n . Then the following are equivalent:(1) u ≤ v in Bruhat order.(2) x u − x v is TNN.(3) x u − x v has (SFL) property.In the next section, we generalize this result as Theorem 5.9.5. Main results
In this section, we give main results as Theorems 5.1, 5.9 and 5.14 with proofs.5.1.
Bigrassmannian statistic.
A bigrassmannian statistic is a meaningful num-ber counting entries of corner sum matrices as the rank function of the finite dis-tributive lattice. We now show a simple and new enumerative formula on entriesof ASMs; this generalizes the author’s formula [12]. the directed graph structureplays a role for a proof.
DIRECTED GRAPH STRUCTURE OF ASMS 19
Theorem 5.1.
For each B ∈ A n , we have β ( B ) = n X i,j =1 ( i − j ) b ij . Proof.
Let α ( B ) be the sum on the right hand side. We will show that β ( B ) = α ( B ) by induction on β ( B ). If β ( B ) = 0, then B = e = ( δ ij ) so that α ( B ) = P ( i − j ) δ ij = 0. Suppose β ( B ) >
0. Choose A ∈ A n such that A → B , say A kl → ij B so that β ( B ) − β ( A ) = | R klij | = ( j − i )( l − k ) . It is now enough to show α ( B ) − α ( A ) = ( j − i )( l − k ), that is, α satisfies the samerecursion (which further shows that α ( B ) is an integer for all B ). Four entries( a ik , a il , a jk , a jl ) must be one of the 16 cases listed in Table 1. It follows, in anycase, that α ( B ) − α ( A ) = n X p,q =1 ( p − q ) b pq − a pq )= − ( i − k ) i − l ) j − k ) − ( j − l ) j − i )( l − k ) . (cid:3) Corollary 5.2. β ( w ) = n X i =1
12 ( i − w ( i )) for w ∈ S n .Proof. Use the theorem. For B = w , we have b ij = 0 if and only if b ij = 1 and j = w ( i ). (cid:3) Example 5.3. β (4312) = 12 (cid:0) (1 − + (2 − + (3 − + (4 − (cid:1) = 9 . ( q TNN) and ( q SFL) properties.
We next introduce a q -analog of (TNN)and (SFL) properties. Motivated by Theorem 5.1, we will consider a q -analog ofour variables x , . . . , x nn . From now on, regard q as a variable taking positivereal numbers so that “ q / ” makes sense. For each ( i, j ), let x ij,q := q ( i − j ) / x ij and call { x ij,q } q -variables . Given a matrix x = ( x ij ), let x q = ( x ij,q ) denote its q -analog. Further, let f ( x q ) mean the polynomial f ( x ,q , . . . , x nn,q ) in x ij and q .In particular, the ASM (Laurent) q -monomial for an ASM A is x Aq := Y i,j ( x ij,q ) a ij (= q β ( A ) x A ) . For example, if A = − , then A q = q / q / − q / q / and x Aq = q x x x − x x . Definition 5.4.
Fix a positive real number q . Say a square matrix A is locallyTNN at q if all minors of A q are nonnegative. Remark 5.5.
Let us make sure that “ A is locally TNN at
1” is equivalent tosaying “ A is TNN” as defined earlier. Definition 5.6.
We say that “ A is q TNN ” if it is locally TNN at q for all q > q -analog of (extended) total nonnegativity. Let g ( x ) be arational function in x , . . . , x nn as before. Definition 5.7.
Say g ( x ) is locally TNN at q if whenever A is locally TNN at q and moreover g ( A q ) is defined, then g ( A q ) ≥
0. Say “ g ( x ) is qTNN ” if it islocally TNN at q for all q > x as ∆ = { ∆ ( x ) , . . . , ∆ m ( x ) } . Definition 5.8.
Say a rational function g ( x ) in x , . . . , x nn has (qSFL) property if there exist F ( x ) , G ( x ) ∈ R [ x ] such that(1) g ( x ) = F ( x ) /G ( x ),(2) F ( x ) = P c i ··· i k ∆ i ( x ) · · · ∆ i k ( x ) with c i ··· i k nonnegative integers, i.e., asubtraction-free polynomial in minors of x ,(3) G ( x ) = Q j ∆ j ( x ) d j with d j nonnegative integers, i.e., a monomial in mi-nors of x and(4) g ( x q ) = F ( x q ) /G ( x q ) ∈ R ( x )[ q ], i.e., g ( x q ) is a polynomial in q .Observe that if g ( x ) and g ( x ) have ( q SFL) property, then so does g + g .5.3. Characterizations of ASM order.Theorem 5.9.
Let
A, B ∈ A n . Then the following are equivalent: (1) A ≤ B in ASM order. (2) x A − x B is q TNN. (3) x A − x B has ( q SFL) property.
We prove (1) = ⇒ (3) = ⇒ (2) = ⇒ (1). Proof. (1) = ⇒ (3): The assertion is obvious for A = B . Let us suppose A A ′ be a locally TNN matrix at q such that G ( A ′ q ) = 0. Then g ( A ′ q ) = F ( A ′ q ) /G ( A ′ q ) ≥ F ( A ′ q ) and each factor in the product G ( A ′ q ) are nonnegative. Thus g ( x ) is locally TNN at q . This is true for all q >
0. Hence g ( x ) is q TNN.(2) = ⇒ (1): This proof is almost same to Drake-Gerrish-Skandera [6, 7]. Nonethe-less, we repeat it here. Suppose A B . We may choose indices k, l ∈ [ n ] such that e A ( k, l ) < e B ( k, l ). Now define the matrix A ′ = ( a ′ ij ) by a ′ ij = ( i ≤ k and j ≤ l A ′ is TNN since all square submatrices of A ′ have determinant0 ,
1, or 2. Now x ij = a ′ ij yields x A − x B (cid:12)(cid:12)(cid:12) x ij = a ′ ij = n Y i,j =1 ( a ′ ij ) a ij − n Y i,j =1 ( a ′ ij ) b ij = Y i ≤ k,j ≤ l a ij − Y i ≤ k,j ≤ l b ij = 2 e A ( k,l ) − e B ( k,l ) < . Thus, x A − x B is not TNN, i.e., x A − x B is not locally TNN at 1. Hence x A − x B is not q TNN. (cid:3)
Corollaries.
We observe several corollaries. First, q = 1 in Theorem 5.9recovers this equivalence: Corollary 5.10.
Let
A, B ∈ A n . Then the following are equivalent: (1) A ≤ B in ASM order. (2) x A − x B is TNN. (3) x A − x B has (SFL) property. Example 5.11.
Let A = − − , B = − − and C = − − . Since A → B → C , x A − x C is TNN and has(SFL) property: x A − x C = ( x A − x B ) + ( x B − x C ) = x x x x x x x (cid:18) x x − x x x x (cid:19) + x x x x x x x (cid:18) x x − x x x x (cid:19) = x x x x x x x x x (cid:18) x x (cid:12)(cid:12)(cid:12)(cid:12) x x x x (cid:12)(cid:12)(cid:12)(cid:12) + x x (cid:12)(cid:12)(cid:12)(cid:12) x x x x (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . As expected, this is a subtraction-free Laurent rational expression in minors of x .It follows that x Aq − x Cq = ( x Aq − x Bq ) + ( x Bq − x Cq )= x x x x x x x x x (cid:18) x x (cid:12)(cid:12)(cid:12)(cid:12) x x x x (cid:12)(cid:12)(cid:12)(cid:12) + x x (cid:12)(cid:12)(cid:12)(cid:12) x x x x (cid:12)(cid:12)(cid:12)(cid:12)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x ij x ij,q . This is a subtraction-free Laurent rational expression in minors of x q ; moreover, β ( A ) = + + + + + = 6 so that x Aq − x Cq = ( q x A − q x B ) + ( q x B − q x C ),certainly a polynomial in q .Here we record some consequence of this example (motivated by recent develop-ments on algebraic combinatorics such as total positivity [9], and cluster algebras[10]); for convenience, we prepare several words. Let us say that a Laurent mono-mial Q i,j x a ij ij is almost positive if a ij ≥ − i, j . Say a minor of a matrix is small if its size is 1 or 2; it is solid if its rows and columns are consecutive. Corollary 5.12. If A < B , then x A − x B has a rational expression as the product L ( x ) × M ( x ) such that L ( x ) is an almost positive Laurent monomial in x , . . . , x nn and M ( x ) is a subtraction-free polynomial in only small solid minors of x (withouta constant term).Proof. By Chain Property, there exists a directed path A → A → A → · · · → A k = B such that β ( A i ) − β ( A i − ) = 1. As seen from Key Lemma, each x A i − x A i +1 is a product of an almost positive Laurent monomial and a subtraction-freepolynomial in only small solid minors without a constant term. Now regarding x A − x B as a sum of such, find its rational expression with choosing a commondenominator. Thus, we obtain the desired expression. (cid:3) DIRECTED GRAPH STRUCTURE OF ASMS 23
Table 2.
Permutation statisticsMahonian Eulerian Bigrassmannianunsigned q -factorial Eulerian polynomial Unknownsigned Wachs [17] D´esarm´enien-Foata [5] Theorem 5.145.5. Signed bigrassmannian statistics.
Permutation statistics is one of im-portant topics in combinatorics on the symmetric groups. In particular,
Maho-nian and
Eulerian are well-known examples (Table 2). More recently, there aresome work on signed Mahonian and signed Eulerian statistics as Wachs [17] andD´esarm´enien-Foata [5]. As one subsequent idea of their work, here we introduce signed bigrassmannian statistics .The inversion number ℓ ( w ) for w ∈ S n is |{ ( i, j ) ∈ [ n ] | i < j and w ( i ) > w ( j ) }| . The sign of w is ( − ℓ ( w ) as often appears in the context of determinants. Now re-call that β ( w ) gives a nonnegative integer |{ v ∈ S n | v ≤ w and v bigrassmannian }| for each permutation w . With these notions, let us introduce a new kind of per-mutation statistics: Definition 5.13.
Define signed bigrassmannian statistics (or signed bigrassman-nian polynomial ) over S n by B n ( q ) = X w ∈ S n ( − ℓ ( w ) q β ( w ) . For example, B ( q ) = 1 , B ( q ) = 1 − q and B ( q ) = 1 − q + 2 q − q (missinga q term; see Figure 3). Theorem 5.14 (Signed bigrassmannian statistics) . For all n ≥ , we have B n ( q ) = n − Y k =1 (1 − q k ) n − k . The idea of our proof is to show the recursion B n ( q ) = B n − ( q ) (1 − q n − ) B n − ( q )(which is not so obvious from the definition of B n ( q )). We derive this equationfrom a series of the lemmas below. Here, we confirm our setting: The notation | | simply denotes the determinant. Let A = ( a ij ) be an n by n matrix with n ≥ Lemma 5.15 (Dodgson’s condensation) . Let A ij denote the submatrix obtained bydeleting i -th row and j -th column from A . Then, we have | A | = | A || A nn | − | A n || A n || A n n | provided | A n n | 6 = 0 .Proof. See Bressoud [2, p.112–113]. (cid:3)
Next, we consider a q -analog of this formula. Lemma 5.16 (a q -analog of Dodgson’s condensation) . With the same notationabove, we have | A q | = | ( A ) q || ( A nn ) q | − q n − | ( A n ) q || ( A n ) q || ( A n n ) q | provided | ( A n n ) q | 6 = 0 .Proof. Apply Dodgson’s condensation to A = A q : | A q | = | ( A q ) || ( A q ) nn | − | ( A q ) n || ( A q ) n || ( A q ) n n | We evaluate these five determinants on the right hand side.(1) | ( A q ) | = | ( q ( i − j ) / a ij ) ni,j =2 | = | ( q ( i − j ) / a i +1 ,j +1 ) n − i,j =1 | = | ( A ) q | .(2) It is similar to show that ( A nn ) q = ( A q ) nn .(3) Using the properties of determinants, we have | ( A q ) n | = | ( q ( i − j ) / a ij ) n,n − i =2 ,j =1 | = | ( q ( i +1 − j ) / a i +1 ,j ) n − i,j =1 | = | ( q (( i − j ) − i − j )+1) / a i +1 ,j ) n − i,j =1 | = q − P i + P j | ( q ( i − j ) / q / a i +1 ,j ) n − i,j =1 | = q ( n − / | ( q ( i − j ) / a i +1 ,j ) n − i,j =1 | = q ( n − / | ( A n ) q | . (4) It is similar to show | ( A q ) n | = q ( n − / | ( A n ) q | by symmetry of rows andcolumns.(5) | ( A q ) n n | = | ( q ( i − j ) / a ij ) n − i,j =2 | = | ( q ( i − j ) / a i +1 ,j +1 ) n − i,j =1 | = | ( A n n ) q | . (cid:3) Lemma 5.17 (determinantal expression) . Consider the matrix A = ( a ij ) with a ij = 1 for all i, j ∈ [ n ] . Then, det( A q ) = B n ( q ) . Moreover, | ( A ) q | = | ( A nn ) q | = | ( A n ) q | = | ( A n ) q | = B n − ( q ) and | ( A n n ) q | = B n − ( q ) . DIRECTED GRAPH STRUCTURE OF ASMS 25
Proof. det( A q ) = P w ∈ S n ( − ℓ ( w ) Q ni =1 q ( i − w ( i )) / = P w ∈ S n ( − ℓ ( w ) q β ( w ) = B n ( q ).In the same way, we can prove the other results for permutation statistics over S n − and S n − . (cid:3) Proof of Theorem 5.14.
Clearly, B ( q ) = 1 and B ( q ) = 1 − q are valid. Suppose n ≥
3. Apply Dodgson’s condensation to A q . With a ij = 1 for all i, j , we get B n ( q ) = B n − ( q ) B n − ( q ) − q ( n − / B n − ( q ) q ( n − / B n − ( q ) B n − ( q )= B n − ( q ) (1 − q n − ) B n − ( q ) . By induction, we conclude that B n ( q ) = B n − ( q ) (1 − q n − ) B n − ( q )= n − Y k =1 (1 − q k ) n − − k ! n − Y k =1 (1 − q k ) n − − k (1 − q n − ) = n − Y k =1 (1 − q k ) n − k . (cid:3) Example 5.18.
Observe that B ( q ) = det q = det q / q / q / q / q / q / = (1 − q ) (1 − q ) = 1 − q + 2 q − q . Let us check B ( q ) = 1 − q + q + 4 q − q − q − q + 4 q + q − q + q . We can see this directly from Table 3. Indeed, our results verify this statistics asfollows: B ( q ) = det q = det q / q / q / q / q / q / q / q / q / q / q / q / = ((1 − q ) (1 − q )) − q (1 − q )= (1 − q ) (1 − q ) (1 − q )= 1 − q + q + 4 q − q − q − q + 4 q + q − q + q . Table 3. signed bigrassmannian statistics over S sign β sign β sign β sign β − − − − − − − − − − − − Concluding remarks
In this article, we introduced a new directed graph structure (
ASM graph ) intoalternating sign matrices. This generalizes Bruhat graph whose edge relation isdefined by transpositions and length functions. The key idea was to considerentries of corner sum matrices rather than entries of ASMs.We established subsequent results of Drake-Gerrish-Skandera [6, 7] on equivalentcharacterizations of Bruhat order in two ways; from permutations to ASMs; q -analogs with respect to the bigrassmannian statistic β . As a by-product, we foundformulas for signed bigrassmannian statistic with a determinantal expression andDodgson’s condensation.We end with several ideas for our subsequent research.(1) Drake-Gerrish-Skandera proved in fact more [6, 7]; Bruhat order is equiv-alent to the monomial nonnegativity (MNN) as well as the Schur nonneg-ativity (SNN). Can we establish some similar results on such propertiesfrom our viewpoints such as ASMs and the q -analog?(2) Recall that we made use of a q -analog of the determinant P w ∈ S n ( − ℓ ( w ) x wq (with x ij = 1) and Dodgson’s condensation to find the signed bigrassman-nian statistic. We next wish to find the unsigned bigrassmannian statis-tic ; Reading [14] originally mentioned this problem. The natural idea isto consider the permanent of the matrix ( q ( i − j ) / ). How can we evaluatethis?(3) Recently, there are many references for research on bivariate permutationstatistics such as Mahonian-Eulerian ; see Skandera [16], for example. Findthe bivariate Mahonian-bigrassmannian statistics P w ∈ S n t ℓ ( w ) q β ( w ) . References [1] T. Ando, Totally positive matrices, Lin. Alg. Appl. 90 (1987) 165-219.
DIRECTED GRAPH STRUCTURE OF ASMS 27 [2] D. Bressoud, Proofs and confirmations, The story of the alternating sign matrix conjecture,Cambridge University Press, Cambridge, 1999. xvi+274 pp.[3] R. Brualdi, L. Deaett, More on the Bruhat order for (0, 1)-matrices, Lin. Alg. Appl. 421(2007), 219-232.[4] R. Brualdi, M. Schroeder, Alternating sign matrices and their Bruhat order, to appear.[5] J. D´esarm´enien, D. Foata, The signed Eulerian numbers, Discrete Math. 99 (1992) 49-58.[6] B. Drake, S. Gerrish, M. Skandera, Monomial nonnegativity and the Bruhat order, Electr.J. Combin. 11 (2006), no. 2, Research Paper 18, 5 pp.[7] B. Drake, S. Gerrish, M. Skandera, Two new criteria for comparison in the Bruhat order,Electr. J. Combin. 11 (2004), no. 1, Note 6, 4 pp.[8] M. Fortin, the MacNeille completion of the poset of partial injective functions, Electr. J.Comb. 15 (2008),
Department of Engineering, Kanagawa University, 3-27-1 Rokkaku-bashi, Yoko-hama 221-8686, Japan.
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