Poset Structure concerning Cylindric Diagrams

The purpose of the present paper is to give a realization of a cylindric diagram as a subset of root systems of type $A_{\kappa-1}^{(1)}$ and several characterization of its poset structure. Furthermore, the set of order ideals of a cylindric diagram is described as a weak Bruhat interval of the Weyl group.


Introduction
A periodic (Young) diagram is a Young diagram consisting of infinitely many cells in Z 2 which is invariant under parallel translations generated by a certain vector ω ∈ Z 2 called the period (see Figure 1).The image of a periodic diagram under the natural projection onto the cylinder Z 2 /Zω is called a cylindric diagram.Diagrams given as a set-difference of two cylindric diagrams are called cylindric skew diagrams.
We note that cylindric skew diagrams have been known to parameterize a certain class of irreducible modules over the Cherednik algebras (double affine Hecke algebras) ( [12,13]) and the (degenerate) affine Hecke algebras ( [1,6]) of type A, where standard tableaux on those diagrams also appear.
The purpose of the present paper is to investigate the poset (θ, ≤) as well as the poset (J (θ), ⊂), where J (θ) denotes the set of cylindric skew diagrams (or proper order ideals) included in θ.We briefly review a description in the classical case.Let λ ⊂ Z 2 be a finite Young diagram.The associated Grassmannian permutation w λ is an element of the Weyl group of the root system R of type A n where n = ♯{c(x) | x ∈ λ}.It is known that the poset (λ, ≤) is dually isomorphic to the poset (R(w −1 λ ), ≤ or ), where R(w −1 λ ) := R + ∩w −1 λ R − and ≤ or is the ordinary order (or the standard order) defined by α≤ or β ⇐⇒ β − α ∈ i∈ [1,n] Z ≧0 α i for α, β ∈ R(w λ ) with Π being the set of simple roots ( [7]).
Let θ be a cylindric diagram in Z 2 /Zω.We would like to describe the poset (θ, ≤) in terms of the root system of type A A key ingredient in our approach is the colored hook length ( [2,4]), given by where H(x) denotes the hook at x and α i are simple roots.(See Section 2.1 for precise definitions.)We will show that the map h embeds the cylindric diagram θ into the set R + of positive (real) roots, and that the image h(θ) is given by the inversion set R(w θ ) associated with a semi-infinite word w θ , which can be thought as an analogue of the Grassmannian permutation.Moreover, we show that the image h(θ) is also characterized as the subset of R + consisting of those elements satisfying where ζ θ is a predominant integral weight determined by θ (see Section 2.2 and 2.3 for details).
Unlike the classical case, the ordinary order in R(w θ ) does not lead a poset isomorphism, and we need to introduce a modified order in R(w θ ) by to obtain a poset isomorphism (θ, ≤) ∼ = (R(w θ ), ), where Π θ is a certain subset of the affine root system (see Section 3.1).
Another description of the poset θ is given by a linear extension or (reverse) standard tableau t on θ, which is by definition a bijective order preserving map θ → Z ≧1 .A linear extension t : θ → Z ≧1 brings a poset structure to Z ≧1 and the resulting poset is an infinite analogue of the heap, which is originally introduced by Stembridge [7].In summary, we have the following: Theorem (Theorem 3.13 and Proposition 3.16).The followings are poset isomorphisms: Another goal of this paper is to describe the poset structure J (θ).For a finite Young diagram λ, it is known that the set J (λ) of order ideals of λ is isomorphic to the interval [e, w λ ] = {u ∈ W | e u w λ } with weak right Bruhat order ([4, Proposition I]).For a cylindric diagram θ, we define a "semi-infinite Bruhat interval" [e, w θ ), and we have the following: Theorem (Theorem 4.6).The map given by Φ(ξ) = w ξ is a poset isomorphism.

Cylindric diagrams as posets
Let (P, ≤) be a poset.For x, y ∈ P , define an interval [x, y] by We say that y covers x if [x, y] = {x, y}.Definition 1.1.Let (P, ≤) be a poset.A subset J of P is called an order filter (resp.order ideal ) if the following condition holds: An order filter (resp.order ideal) J is said to be proper if J = P , and it is said to be non-trivial if J = P nor J = ∅.
For ω ∈ Z ≧1 × Z ≦−1 , we let Zω denote the subgroup of (the additive group) Z 2 generated by ω, and define the cylinder C ω by For x, y ∈ C ω , write x ≤ y if there exists x, ỹ ∈ Z 2 such that π(x) = x, π(ỹ) = y and x ≤ ỹ.It is not difficult to see the following: Then the relation ≤ on C ω is a partial order, and the projection π is order preserving.
In the rest of this section, we fix and λ is a fundamental domain of λ with respect to the action of Z(m, −ℓ).
If λ ∈ P ω then λ is a periodic diagram of period ω and λ is a cylindric diagram.Moreover, any periodic (resp.cylindric) diagram of period ω is of the form λ (resp.λ) for some λ ∈ P ω .
For a poset P and its order filter J, we denote the set-difference P \ J also by P/J.It is easy to see the following: Proposition 1.6.For a subset ξ of C ω , the following conditions are equivalent : (i) ξ is a proper order ideal of a cylindric diagram in C ω .(ii) ξ is a set-difference θ/η of two cylindric diagrams θ, η in C ω with θ ⊃ η. (iii) ξ is an intersection of a proper order ideal and a proper order filter of C ω .(iv) ξ is a finite subset of C ω and satisfies the following condition: (v) ξ is a finite subset of C ω and satisfies the following condition: We denote the set of proper order ideals of θ by J (θ) and regard it as a poset with the inclusion relation.Note that any ξ ∈ J (θ) is a finite set and thus J (θ) = ∞ n=0 J n (θ), where we put

Standard tableaux
In the rest of present section, fix a cylindric diagram θ in C ω .
We denote by ST(θ) the set of standard tableaux of θ.
(2) For a finite poset P with |P | = n, a standard tableau of P is a bijection t : P → [1, n] satisfying x < y =⇒ t(x) < t(y).
We denote by ST(P ) the set of standard tableaux of P .Remark 1.9.Our standard tableaux are usually referred to as reverse standard tableaux.♦ Let t ∈ ST(θ).It is easy to see that the subset t −1 ([1, n]) of θ is a proper order ideal, and moreover the restriction t| t −1 ([1,n]) is a standard tableau on t −1 ([1, n]).Conversely, for ξ ∈ J n (θ), any standard tableau on ξ can be extended to a standard tableau on θ.In summary, we have the following: Moreover, for each t ∈ ST(θ), the restriction t → t| t −1 ([1,n]) gives a surjective map

Content map and bottom set
Let Θ be a periodic diagram of period ω.Define the content map which we denote by the same symbol c.It is easy to show the following: Proposition 1.11.For x, y ∈ θ, the followings hold: (1) If c(x) − c(y) ≡ 0, ±1 mod κ, then x and y are comparable.
Let i ∈ Z/κZ.By Proposition 1.11 (1), the inverse image c −1 (i) is non-empty totally ordered subset of θ.Let b i denote the minimum element in c −1 (i).Let κ ∈ Z ≧2 .In the rest, we often identify Z/κZ with {0, 1, . . ., κ − 1}.Let h be a (κ + 1)dimensional vector space and choose elements α ∨ i (i ∈ Z/κZ) and d of h so that forms a basis for h.Let h * be the dual space of h.Define elements α j (j ∈ Z/κZ) and ̟ 0 of h * by where •, • : h * × h → Z is the natural pairing and the integer a ij is defined by The weights ̟ 0 , ̟ 1 , . . ., ̟ κ−1 are called fundamental weights.Put , which is called the null root (resp.the null coroot).For i ∈ Z/κZ, define the simple reflection s i ∈ GL(h * ) by Define the affine Weyl group W of type A κ−1 as the subgroup of GL(h * ) generated by simple reflections: The following is well-known: Proposition 1.14.The group W has the following fundamental relations: For w ∈ W , we define the length ℓ(w) of w as the smallest r for which an expression (or a word) Define the action of W on h by We put The set Π (resp.Π ∨ ) is called the set of simple roots (resp.the set of simple coroots), and Then R (resp.R ∨ ) is the set of real roots (resp.coroots) and R ⊔Zδ is the affine root system.Define the set R + of positive (real) roots and the set R − of negative (real) roots by For i, j ∈ Z with i < j, we define where k = k mod κZ ∈ Z/κZ.The followings are well-known: From the description of R above, the following two lemmas follow easily and they will be used later: 2 Hooks in cylindric diagrams

Colored hook length
In this section, we will introduce colored hook length, which is a key ingredient in this paper.
In the rest of this paper, we use the following notations: α(y).
We call h(x) the colored hook length at x.
(2) A conjectural hook formula concerning the number of standard tableaux on cylindric skew diagrams is proposed in [11], where the hook length at x ∈ θ is given by |H Let Γ = {b i | i ∈ Z/κZ} be the bottom set of θ, where b i is the minimum element of c −1 (i) as before.
For α = i∈Z/κZ c i α i ∈ Q + , define its support by For example, we have Supp(δ) = Γ.Let x ∈ θ with N(h(x)) = 0. Then Supp(h(x)) is a non-empty, proper and connected subset of Γ.

Weyl group elements and their inversion sets
The following proposition gives an alternative expression for h(x).Proposition 2.9.For any x ∈ θ and t ∈ ST(θ), it holds that where n = t(x).
The proof of Proposition 2.9 will be given in the next section.In the rest of this section, we will see some consequences of the proposition.
Let t ∈ ST(θ).For n ∈ Z ≧1 , we define an element w θ,t [n] of W by and we set w θ,t [0] = e.
Proof.Put p k = t −1 (k) for k ≧ 1.By Proposition 2.9 and Theorem 2.8, we have Therefore we have ℓ(w θ,t [n]) = n and thus the expression (2.4) is reduced.
By (2.3), (2.4) and Proposition 2.11, we obtain the following proposition: Proposition 2.12.Let t ∈ ST(θ) and n ∈ Z ≧1 .Then it holds that In particular, it holds that R(w In particular, R(w θ,t ) is independent of t and we will denote it just by R(w θ ) in the rest.
Remark 2.13.The set R(w θ ) can be thought as the "inversion set" associated with the semi-infinite word (2) An element w of W is said to be ζ-minuscule if

Proof of Proposition 2.9
For t ∈ ST(θ) and x ∈ θ, we put where n = t(x).For x ∈ θ, put x S = x + (1, 0), x E = x + (0, 1), x SE = x + (1, 1) ∈ C ω .We will use the following lemma later: (2.7) (2.8) (2.9) (2.10) where Actually, if c(p d ) − r = 0, ±1 then p d is comparable with p j and p i , and hence p j > p d > p i .But such d must be k 1 or k 2 .Now we have (2) Suppose that x S , x E / ∈ θ, or equivalently, suppose that x is minimal element in Γ.Let x = p j .Then p d (d ∈ [1, j − 1]) is not comparable with p j .Hence The other cases are reduced to the case where x is minimal in Γ, via a similar argument as in the proof of the statement (1), Proof of Proposition 2.9.Let x ∈ θ.Put x S = x + (1, 0), x E = x + (0, 1), x SE = x + (1, 1).It is easy to see the following: On the other hand, we have shown that γ t (x) satisfies the same recurrence relations in Lemma 2.18.
3 Poset structure of cylindric diagrams

Partial orders on the inversion set
Recall that Q denote the root lattice: Q = i∈Z/κZ Zα i .
Definition 3.1.Define the partial order ≤ or on Q by The order ≤ or is called the ordinary order.
The restriction of the ordinary order defines a poset structure on R(w θ ).
We will introduce a modified ordinary order , for which we will have (θ, ≤) ∼ = (R(w θ ), ).Let Γ = {b i | i ∈ Z/κZ} be the bottom set of θ, where b i is the element such that c(b i ) = i.Let Γ max (resp.Γ min ) denote the set of maximal (resp.minimal) elements in Γ.

Poset isomorphism
Our next goal is to prove that the order preserving map is actually a poset isomorphism.We start with some preparations.
As before, we denote by Supp(α) the support of α ∈ Q.The following lemma is almost obvious from Definition 3.3.
It is easy to see the next lemma: Proof.
In particular, if x and y are incomparable, then h(x) and h(y) are also incomparable with respect to ≤ or .Proof.By Corollary 3.8, we have x < y =⇒ h(x)≤ or h(y).
(1) Suppose that N(y) − N(x) = 1.In this case, By definition, h(x 0 ) and h(y 0 ) are positive roots.By Lemma 1.15, h(x 0 ) − δ is also a root and it is not positive.Therefore δ − h(x 0 ) ∈ R + and h(y) − h(x) is a sum of two positive roots.This implies h(x)≤ or h(y).Combining with (3.3), we have x − δ≤ or y≤ or x.
(3) Suppose that N(y) − N(x) = 0. Note that x 0 and y 0 are incomparable this case, and it follows from Lemma 3.11 that h(y 0 ) and h(x 0 ) are also incomparable with respect to ≤ or .Now we have and hence h(x) and h(y) are incomparable with respect to ≤ or .
x is a poset isomorphism.
Proof.By Proposition 3.7, we have Thus the statement follows if we prove that x and y are incomparable =⇒ h(x) and h(y) are incomparable with respect to .
Suppose that x and y are incomparable.Then, putting n = N(h(x)), we have N(h(y)) = n + 1, n or n − 1 by Lemma 3.12.First we assume that N(y) = n.Then Lemma 3.12 implies that h(x) and h(y) must be incomparable.
Thus there exist i 1 , . . ., i s for which we have h(y 0 ) = β i 1 + • • • + β is , but this contradics (3.4).Therefore h(y) − h(x) = h(y 0 ) + (δ − h(x 0 )) cannot be a sum of elements in Π θ , and thus h(x) and h(y) are incomparable with respect to .The same argument implies that h(x) and h(y) are incomparable also in the case where N(h(y)) = n − 1.
Proposition 3.14.Let α, β ∈ R(w θ ) with α β.Then there exists a sequence In other words, the partial order on R(w θ ) coincides with the transitive closure of the relations α • β whenever β − α ∈ Π θ . (3.6) Proof.Let tc denote the transitive closure of the relations above.It follows from the same argument in the proof of Proposition 3.7 that x ≤ y =⇒ h(x) tc h(y) for any x, y ∈ θ.It is clear that Combining with Theorem 3.13, the statement follows.

Heaps
Let θ be a cylindric diagram.Recall that standard tableaux on θ have been defined as order preserving bijection from (θ, ≤) to (Z ≧1 , ≦).Through the bijection t, the set Z ≧1 inherits a partial order from θ, which we will investigate in this section.where Proposition 3.16.Let θ be a cylindric diagram and t a standard tableau on θ.Then, the map t : θ → Z ≧1 gives a poset isomorphism Proof.Let x, y ∈ θ.Suppose that x < y is a covering relation in θ.Then y = x − (1, 0) or y = x − (0, 1) and it is easy to see that t(x) < t(y) and s(x)s(y) = s(y)s(x).Hence t(x) t t(y).
Conversely, suppose that t(x) ≺ t t(y) is a covering relation in Z ≧1 .Then s(x)s(y) = s(y)s(x) or c(x) = c(y), and hence c(x) − c(y) = 0, ±1.By Proposition 1.11 (1), x and y are comparable.Since t is order preserving, we must have x < y, and hence h is a poset isomorphism.
The posets (Z ≧1 , t ) are thought as semi-infinite analogue of heaps introduced by Stembridge [8].Stembridge also introduced the heap order on the inversion sets.We treat a slightly modified version of heap order by Nakada [2].Definition 3.17.Define a partial order ≤ hp on R(w θ ) as the transitive closure of the relations α ≤ hp β whenever α≤ or β and α, β ∨ = 0. Proposition 3.18.The map h : θ → R(w θ ) gives a poset isomorphism In other words, the partial order ≤ hp and on R(w θ ) coincide.
Proof.Let x, y ∈ θ.Suppose that x < y is a covering relation in θ.Then h(x)≤ or h(y) and h If h(y), h(x) ∨ = 0 then h(y) − h(x) ≡ h(y) mod Zδ by Lemma 1.16, and thus h(x) = kδ for some k ∈ Z.This is a contradiction.Therefore h(x), h(y) ∨ = 0, from which it follows that h(x) ≤ hp h(y).
Next, suppose that h(x) ≤ hp h(y) is a covering relation.Put x 0 = x + N(h(x))(0, ℓ) and by assumption.We assume that x and y are incomparable.Then as h(x)≤ or h(y), we have N(h(y)) = N(h(x)) + 1 and h(y 0 )≤ or h(x 0 )≤ or δ (3.8) by Lemma 3.12.Moreover, by (3.5) in the proof of Theorem 3.13, we have (See also Figure 8.) Recall that positive roots h(x 0 ) and h(y 0 ) can be expressed as h(x 0 ) = α ij and h(y 0 ) = α kl for some i, j, k, l ∈ Z with i < j, k < l.By (3.8) and (3.9), the indices i, j, k and l can be chosen in such a way that they satisfy j − i ≦ κ − 1 and i < k < l < j.Thus we have This contradicts (3.7).Therefore x and y are comparable, and thus x < y as h(x) < h(y).
4 Poset structure of the set of order ideals

Standard tableaux on cylindric skew diagrams
For a poset P , let J (P ) denote the set of proper order ideals and regard J (P ) as a poset with the inclusion relation.
Let ω ∈ Z ≧1 ×Z ≦−1 and fix a cylindric diagram θ in C ω .In this section, we will investigate the poset structure of the set J (θ) of order ideals of θ, in other words, cylindric skew diagrams included in θ.
Recall that any cylindric skew diagram ξ ∈ J (θ) is a finite set and For ξ ∈ J n (θ) and t ∈ ST(ξ), define a word w ξ,t by We sometimes regard w ξ,t as a Weyl group element.
Proposition 4.1.The word w ξ,t is reduced.As an element of Weyl group, w ξ,t is fully commutative and independent of t.
Proof.It follows from Lemma 1.10 that the standard tableau t on ξ can be extended to a standard tableau t on θ, for which we have w θ, t[n] = w ξ,t .By Proposition 2.11 and Proposition 2.17, the right hand side of (4.1) is a reduced expression and w ξ,t is a fully commutative element of W .It follows from Proposition 2.12 that R(w ξ,t ) = {h(x) | x ∈ ξ}.
Hence the set R(w ξ,t ) is independent of t and so is w ξ,t .
We denote by w ξ the Weyl group element determined by the word w ξ,t for a/any standard tableau t ∈ ST(ξ).gives a bijection from ST(ξ) to the set of reduced expressions for w ξ .
where W and R denote the Weyl group and the root system of type A κ−1 respectively.We will see the relation between the results above and preceding works.Let n ∈ Z ≧1 and λ be a partition of n.Fix t ∈ ST(λ/φ) and put w λ := w λ/φ = s(t −1 (n))s(t −1 (n − 1)) • • • s(t −1 (1)).
The element w λ is independent of t and it is called the Grassmannian permutation associated with λ.
Let π : Z 2 → C ω be the natural projection.The cylinder C ω inherits a Z 2 -module structure via π.Define a poset structure on Z 2 by (a, b) ≤ (a ′ , b ′ ) ⇐⇒ a ≧ a ′ and b ≧ b ′ as integers.

Figure 1 Definition 1 . 5 .
Figure 1 indicates a periodic diagram of period ω = (4, −5).The set consisting of colored cells is a fundamental domain with respect to the action of Zω, and it is in one to one correspondence with the associated cylindric diagram.Definition 1.5.Let m, ℓ ∈ Z ≧1 .A non-increasing sequence λ = (λ 1 , . . ., λ m ) of (possibly negative) integers is called a generalized partition of length m.For ω = (m, −ℓ) ∈ Z ≧1 ×Z ≦−1 , we denote by P ω the set of generalized partitions of length m satisfying

Figure 5 :
Figure 5: The sets Arm(x) and Leg(x) for x in the cylindric diagram.

Figure 7 :
Figure 7: The cells in the shadow are incomparable with x = x n .

Definition 3 .
15. Let t ∈ ST(θ).Define a partial order t on Z ≧1 as the transitive closure of the relations a t b whenever a ≦ b and either s ia s i b = s i b s ia or i a = i b .