Equivariant K -theory and hook formula for skew shape on d-complete set

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1 Equivariant K -theory and hook formula for skew shape on d-complete set Hiroshi Naruse Graduate School of Education University of Yamanashi Algebraic and Enumerative Combinatorics in Okayama 2018/02/20

2 Introduction Main Theorem Idea of Proof Outline Equivariant K -theory Kashiwara thick flag variety G/P Equivariant K -theory K T (G/P ) and localization map Littlewood-Richardson coefficients Chevalley formula Application to d-complete posets Setup Billey type formula Chevalley formula for d-complete poset Example Birds

3 Introduction Main Theorem Idea of Proof Outline Equivariant K -theory Kashiwara thick flag variety G/P Equivariant K -theory K T (G/P ) and localization map Littlewood-Richardson coefficients Chevalley formula Application to d-complete posets Setup Billey type formula Chevalley formula for d-complete poset Example Birds

4 Main Theorem (N-Okada) Theorem P: a connected d-complete poset, c : P I coloring. F : an order filter of P, A(P \ F) := {σ : P \ F N p q = σ(p) σ(q)}. Then we have z σ = v B(D) z[h P(v)] v P\D (1 z[h P(v)]), (1) where z σ = σ A(P\F ) v P\F z σ(v) c(v), D E P (F ) z[h P (v)] is the hook monomial, E P (F) is the set of excited giagrams of F in P, and B(D) is the set of excited peaks of D in P.

5 Introduction Main Theorem Idea of Proof Outline Equivariant K -theory Kashiwara thick flag variety G/P Equivariant K -theory K T (G/P ) and localization map Littlewood-Richardson coefficients Chevalley formula Application to d-complete posets Setup Billey type formula Chevalley formula for d-complete poset Example Birds

6 Idea of Proof Let G P\F be the left hand side and E P\F be the right hand side [ ξ w F wp of the equation (1). Let Z P\F := ξ w P wp. For order filters, ]e α i =z i F F means F F and F \F is non-empty antichain. 1) From Billey type formula we can see Z P\F = E P\F. 2) By Chevalley formula we have a recurrence relation on Z P\F. Z P\F = 1 1 z[p\f] ( 1) F \F 1 Z P\F. F F 3) By inclusion-exclusion principle we can easily deduce the same recurrence relation on G P\F. G P\F = 1 1 z[p\f] ( 1) F \F 1 G P\F. F F Comparing initial values we get G P\F = Z P\F.

7 Idea of Proof Let G P\F be the left hand side and E P\F be the right hand side [ ξ w F wp of the equation (1). Let Z P\F := ξ w P wp. For order filters, ]e α i =z i F F means F F and F \F is non-empty antichain. 1) From Billey type formula we can see Z P\F = E P\F. 2) By Chevalley formula we have a recurrence relation on Z P\F. Z P\F = 1 1 z[p\f] ( 1) F \F 1 Z P\F. F F 3) By inclusion-exclusion principle we can easily deduce the same recurrence relation on G P\F. G P\F = 1 1 z[p\f] ( 1) F \F 1 G P\F. F F Comparing initial values we get G P\F = Z P\F.

8 Idea of Proof Let G P\F be the left hand side and E P\F be the right hand side [ ξ w F wp of the equation (1). Let Z P\F := ξ w P wp. For order filters, ]e α i =z i F F means F F and F \F is non-empty antichain. 1) From Billey type formula we can see Z P\F = E P\F. 2) By Chevalley formula we have a recurrence relation on Z P\F. Z P\F = 1 1 z[p\f] ( 1) F \F 1 Z P\F. F F 3) By inclusion-exclusion principle we can easily deduce the same recurrence relation on G P\F. G P\F = 1 1 z[p\f] ( 1) F \F 1 G P\F. F F Comparing initial values we get G P\F = Z P\F.

9 Idea of Proof Let G P\F be the left hand side and E P\F be the right hand side [ ξ w F wp of the equation (1). Let Z P\F := ξ w P wp. For order filters, ]e α i =z i F F means F F and F \F is non-empty antichain. 1) From Billey type formula we can see Z P\F = E P\F. 2) By Chevalley formula we have a recurrence relation on Z P\F. Z P\F = 1 1 z[p\f] ( 1) F \F 1 Z P\F. F F 3) By inclusion-exclusion principle we can easily deduce the same recurrence relation on G P\F. G P\F = 1 1 z[p\f] ( 1) F \F 1 G P\F. F F Comparing initial values we get G P\F = Z P\F.

10 Idea of Proof Let G P\F be the left hand side and E P\F be the right hand side [ ξ w F wp of the equation (1). Let Z P\F := ξ w P wp. For order filters, ]e α i =z i F F means F F and F \F is non-empty antichain. 1) From Billey type formula we can see Z P\F = E P\F. 2) By Chevalley formula we have a recurrence relation on Z P\F. Z P\F = 1 1 z[p\f] ( 1) F \F 1 Z P\F. F F 3) By inclusion-exclusion principle we can easily deduce the same recurrence relation on G P\F. G P\F = 1 1 z[p\f] ( 1) F \F 1 G P\F. F F Comparing initial values we get G P\F = Z P\F.

11 Idea of Proof Let G P\F be the left hand side and E P\F be the right hand side [ ξ w F wp of the equation (1). Let Z P\F := ξ w P wp. For order filters, ]e α i =z i F F means F F and F \F is non-empty antichain. 1) From Billey type formula we can see Z P\F = E P\F. 2) By Chevalley formula we have a recurrence relation on Z P\F. Z P\F = 1 1 z[p\f] ( 1) F \F 1 Z P\F. F F 3) By inclusion-exclusion principle we can easily deduce the same recurrence relation on G P\F. G P\F = 1 1 z[p\f] ( 1) F \F 1 G P\F. F F Comparing initial values we get G P\F = Z P\F.

12 Introduction Main Theorem Idea of Proof Outline Equivariant K -theory Kashiwara thick flag variety G/P Equivariant K -theory K T (G/P ) and localization map Littlewood-Richardson coefficients Chevalley formula Application to d-complete posets Setup Billey type formula Chevalley formula for d-complete poset Example Birds

13 Kashiwara thick flag variety G/P A = (a ij ) i,j I a symmetrizable generalized Cartan matrix, Γ the corresponding Dynkin diagram with node set I. Then the associated Kac Moody group G over C is constructed from the following data: the weight lattice Z-module Λ Z m, the (linearly independent) simple roots Π = {α i : i I} Λ, the simple coroots Π = {αi : i I} Λ, the fundamental weights {λ i : i I} Λ where Λ = Hom Z (Λ, Z) is the dual lattice. We will write the canonical pairing, : Λ Λ Z. These satisfy α i, α j = a ij, α i, λ j = δ ij.

14 The Weyl group W is generated by simple reflections s i (i I) and it is known to be a crystallographic Coxeter group. Let B be a Borel subgroup and B the opposite Borel i.e. B B = T is the maximal torus. Let us fix a subset J I. Then we can consider the subgroup W J W generated by s j, (j J) and the parabolic subgroup P B corresponding to J. Let W J = W /W J be the set of minimum length coset representatives. Then we can consider the Kashiwara thick partial flag variety X P = G/P.

15 X P = G/P has cell decomposition. X P = w W J X w where X w = BwP /P is the B-orbit of T -fixed point corresponding to w, e w = wp /P X P. The Zariski closure X w = X w of X w is called the Schubert variety. Using Bruhat order on W J induced from W, we have cell decomposition X w = u W J,u w X u. X w has codimension l(w) in X P and it is infinite dimensional if W =.

16 Introduction Main Theorem Idea of Proof Outline Equivariant K -theory Kashiwara thick flag variety G/P Equivariant K -theory K T (G/P ) and localization map Littlewood-Richardson coefficients Chevalley formula Application to d-complete posets Setup Billey type formula Chevalley formula for d-complete poset Example Birds

17 Equivariant K -theory K T (G/P ) We can consider T -equivariant K -theory of X P = G/P K T (G/P ) is the Grothendieck group of coherent sheaves on G/P. It has ring structure and K T (X P ) = w W J K T (pt)[o w ], where O w is the structure sheaf of the Schubert variety X w and [O w ] is the corresponding class in K T (X P ). K T (pt) is the T -equivariant K -theory of a point and it can be identified with the representation ring of T. K T (pt) R(T ) Z[Λ]. Each element in Z[Λ] is expressed as a (Z-)linear combination of e λ (λ Λ). e λ corresponds to the class of line bundle L λ with character λ.

18 Schubert class and Localization map For w W J, we will write ξ w = [O w ] K T (X P ) and call it T -equivariant Schubert class. Each v W J gives a T -fixed point e v = vp /P X. Then the inclusion map ι v : {e v } X induces the pull-back ring homomorphism, called the localization map at v, ι v : K T (X P ) K T (e v ) = Z[Λ]. For two elements v, w W J, we denote by ξ w v the image of ξ w under the localization map ι v: ξ w v = ι v([o w ]).

19 Billey type formula for ξ w v (Lam-Schilling-Shimozono 2010) Proposition Let v, w W J, and fix a reduced expression v = s i1 s i2... s in of v. Then we have ξ w v = (k 1,...,k r ) ( 1) r l(w) r a=1 ( 1 e β(k a) ), (2) where the summation is taken over all sequences (k 1,..., k r ) such that 1 k 1 < k 2 < < k r N and s ik1 s ik r = w (with respect to the Demazure product), and β (k) is given by β (k) = s i1... s ik 1 (α ik ) for 1 k N.

20 Introduction Main Theorem Idea of Proof Outline Equivariant K -theory Kashiwara thick flag variety G/P Equivariant K -theory K T (G/P ) and localization map Littlewood-Richardson coefficients Chevalley formula Application to d-complete posets Setup Billey type formula Chevalley formula for d-complete poset Example Birds

21 Littlewood-Richardson coefficients We consider the structure constants for the multiplication in K T (X ) with respect to the Schubert classes. [O u ][O v ] = where u, v, w W J, c w u,v K T (pt). Lemma If c w u,v 0, then u w and v w. Lemma For u, w W J, we have w W J c w u,v[o w ], c w u,w = ξ u w. (3)

22 Littlewood-Richardson coefficients We consider the structure constants for the multiplication in K T (X ) with respect to the Schubert classes. [O u ][O v ] = where u, v, w W J, c w u,v K T (pt). Lemma If c w u,v 0, then u w and v w. Lemma For u, w W J, we have w W J c w u,v[o w ], c w u,w = ξ u w. (3)

23 Littlewood-Richardson coefficients We consider the structure constants for the multiplication in K T (X ) with respect to the Schubert classes. [O u ][O v ] = where u, v, w W J, c w u,v K T (pt). Lemma If c w u,v 0, then u w and v w. Lemma For u, w W J, we have w W J c w u,v[o w ], c w u,w = ξ u w. (3)

24 Lemma Let u, v, w W J and s W J a simple reflection. If cs,w w cs,u, u then we have cu,v w = 1 cs,w w cs,u u cs,uc x x,v w cs,yc w u,v y. u<x w u y<w In particular, we have c w u,w = 1 cs,w w cs,u u u<x w c x s,uc w x,w. i.e. ξ u w = 1 cs,w w cs,u u cs,uξ x x w. (4) u<x w

25 Proof. Consider the associativity ([O s ][O u ]) [O v ] = [O s ] ([O u ][O v ]) and take the coefficients of [O w ] in the both hand sides. c u s,uc w u,v + u<x w then we get cu,v w 1 = cs,w w cs,u u c x s,uc w x,v = c w s,wc w u,v + u<x w c x s,uc w x,v u y<w u y<w c w s,yc y u,v, c w s,yc y u,v.

26 Introduction Main Theorem Idea of Proof Outline Equivariant K -theory Kashiwara thick flag variety G/P Equivariant K -theory K T (G/P ) and localization map Littlewood-Richardson coefficients Chevalley formula Application to d-complete posets Setup Billey type formula Chevalley formula for d-complete poset Example Birds

27 T -quivariant Chevalley formula W -orbit of the set of simple roots Π(resp. simple coroots Π ) determine the set of real roots (resp. real coroots) Φ = W Π (resp. Φ = W Π ) and the decomposition of Φ (resp. Φ ) into the positive system Φ + (resp. Φ +) and the negative system Φ (resp. Φ ). For a dominant weight λ Λ, we define H λ = {(γ, k) : γ Φ +, 0 k < γ, λ, k N}. Fix a total order on the Dynkin node I so that I = {i 1 < < i r }. define a map ι : H λ Q r+1 by r ι c i αi j, k 1 = γ, λ (k, c 1,, c r ). j=1 Then it is known that ι is injective.

28 We define a total ordering < on H λ by h < h ι(h) < lex ι(h ), where < lex is the lexicographical ordering on Q r+1. For h = (γ, k), we define affine transformations r h and r h on Λ by r h (µ) = µ γ, µ γ, r h (µ) = r h (µ) + ( γ, λ k ) γ. Note that r h = s γ is the reflection corresponding to the positive root γ.

29 Lenart-Shimozono (2014) Proposition Let s = s i W J be a simple reflection. Then for w, v W J we have 1 e λ i vλ i if w = v, cs,v w = ( 1) r 1 e λ i v r h1 r h r λ i if w > v, (5) (h 1,,h r ) where the summation is taken over all sequences (h 1,, h r ) of length r 1 satisfying the following two conditions: (H1) h 1 > h 2 > > h r in H λi, (H2) v vr h1 vr h1 r h2 vr h1 r hr = w is a saturated chain in W J.

30 Introduction Main Theorem Idea of Proof Outline Equivariant K -theory Kashiwara thick flag variety G/P Equivariant K -theory K T (G/P ) and localization map Littlewood-Richardson coefficients Chevalley formula Application to d-complete posets Setup Billey type formula Chevalley formula for d-complete poset Example Birds

31 Application to d-complete posets In what follows, let P be a connected d-complete poset with top tree Γ together with d-complete coloring c : P I. α P the simple root, λ P the fundamental weight corresponding to the color i P of the maximum element of P. We apply the above argument to the Kashiwara thick partial flag variety X P = G/P, where P is the maximal parabolic subgroup corresponding to J = I \ {i P }. In this case, the parabolic subgroup W J coincides with the stabilizer of λ P in W, and the minimum length coset representatives W J is denoted by W λ P. For p P, we put α(p) = α c(p), α (p) = α c(p), s(p) = s c(p).

32 Take a linear extension and label the elements of P as p 1,, p N (N = #P) so that p i < p j in P implies i < j. Then we constructs an element w = w P W by putting w P = s(p 1 )s(p 2 ) s(p N ). For an order filter F = {p i1,, p ir } (i 1 < < i r ), we define F F If p = p k P, then we define Proposition (Proctor 2014) w F = s(p i1 ) s(p ir ). w F w F in Bruhat order β(p k ) = s(p 1 ) s(p k 1 )α(p k ), z[h P (p)] = [e β(p) ] e α i =zi.

33 Take a linear extension and label the elements of P as p 1,, p N (N = #P) so that p i < p j in P implies i < j. Then we constructs an element w = w P W by putting w P = s(p 1 )s(p 2 ) s(p N ). For an order filter F = {p i1,, p ir } (i 1 < < i r ), we define F F If p = p k P, then we define Proposition (Proctor 2014) w F = s(p i1 ) s(p ir ). w F w F in Bruhat order β(p k ) = s(p 1 ) s(p k 1 )α(p k ), z[h P (p)] = [e β(p) ] e α i =zi.

34 Take a linear extension and label the elements of P as p 1,, p N (N = #P) so that p i < p j in P implies i < j. Then we constructs an element w = w P W by putting w P = s(p 1 )s(p 2 ) s(p N ). For an order filter F = {p i1,, p ir } (i 1 < < i r ), we define F F If p = p k P, then we define Proposition (Proctor 2014) w F = s(p i1 ) s(p ir ). w F w F in Bruhat order β(p k ) = s(p 1 ) s(p k 1 )α(p k ), z[h P (p)] = [e β(p) ] e α i =zi.

35 Take a linear extension and label the elements of P as p 1,, p N (N = #P) so that p i < p j in P implies i < j. Then we constructs an element w = w P W by putting w P = s(p 1 )s(p 2 ) s(p N ). For an order filter F = {p i1,, p ir } (i 1 < < i r ), we define F F If p = p k P, then we define Proposition (Proctor 2014) w F = s(p i1 ) s(p ir ). w F w F in Bruhat order β(p k ) = s(p 1 ) s(p k 1 )α(p k ), z[h P (p)] = [e β(p) ] e α i =zi.

36 Take a linear extension and label the elements of P as p 1,, p N (N = #P) so that p i < p j in P implies i < j. Then we constructs an element w = w P W by putting w P = s(p 1 )s(p 2 ) s(p N ). For an order filter F = {p i1,, p ir } (i 1 < < i r ), we define F F If p = p k P, then we define Proposition (Proctor 2014) w F = s(p i1 ) s(p ir ). w F w F in Bruhat order β(p k ) = s(p 1 ) s(p k 1 )α(p k ), z[h P (p)] = [e β(p) ] e α i =zi.

37 Given a subset D = {p i1,, p ir } (i 1 < < i r ) of P, we define elements w D W and wd W by putting w D = s(p i1 )s(p i2 ) s(p ir ) and w D = s(p i 1 ) s(p i2 ) s(p ir ), where is the Demazure product. Proposition Let F be an order filter of P and D P. (1) D E P (F) w D = w F and E = F (2) D E P (F) w D = w F For the case (2) D is uniquely expressed as D = D S s.t. D E P (F), S B(D ).

38 Introduction Main Theorem Idea of Proof Outline Equivariant K -theory Kashiwara thick flag variety G/P Equivariant K -theory K T (G/P ) and localization map Littlewood-Richardson coefficients Chevalley formula Application to d-complete posets Setup Billey type formula Chevalley formula for d-complete poset Example Birds

39 Proposition We have ξ w F wp = Billey type formula E E P (F ) ( 1) #E #F (1 z[h P (p)]), (6) p E under the identification z i = e α i (i I). We can rewrite the above expression as ξ w F wp = (1 z[h P (p)]) z[h P (p)]. D E P (F ) p D ξ w F wp ξ w = P wp D E P (F ) p B(D) v B(D) z[h P(v)] v P\D (1 z[h P(v)])

40 Introduction Main Theorem Idea of Proof Outline Equivariant K -theory Kashiwara thick flag variety G/P Equivariant K -theory K T (G/P ) and localization map Littlewood-Richardson coefficients Chevalley formula Application to d-complete posets Setup Billey type formula Chevalley formula for d-complete poset Example Birds

41 Chevalley formula for d-complete poset Proposition Let P be a connected d-complete poset and s = s ip. For two order filters F and F of P, we have 1 z[f ] if F = F, c w F s,w F = ( 1) #(F \F ) 1 z[f] if F F and F \ F is an antichain, 0 otherwise, (7) under the identification z i = e α i (i I).

42 Lemma Let F be an order filter of P and h = (γ, k) H λp. If w F r h W λ P and w F w F r h w P, then there exists p P such that F {p} is an order filter of P, w F r h = w F and γ = γ (p). In this case k = 0, because γ, λ P = 1, and r h λ P = λ P.

43 To deduce recurrence relation on Z P\F := ξw F wp ξ w P wp, recall ξ u w = 1 cs,w w cs,u u cs,uξ x x w. (8) u<x w Change variables u = w F,w = w P, x = w F. Then we have ξ w F wp = Note that c w F s,w F Z P\F = z[f] z[f] z[p] = 1 z[f]. 1 1 z[p\f] ( 1) F \F 1 ξ w F wp. (9) F F ( 1) F \F 1 Z P\F F F

44 To deduce recurrence relation on Z P\F := ξw F wp ξ w P wp, recall ξ u w = 1 cs,w w cs,u u cs,uξ x x w. (8) u<x w Change variables u = w F,w = w P, x = w F. Then we have ξ w F wp = Note that c w F s,w F Z P\F = z[f] z[f] z[p] = 1 z[f]. 1 1 z[p\f] ( 1) F \F 1 ξ w F wp. (9) F F ( 1) F \F 1 Z P\F F F

45 To deduce recurrence relation on Z P\F := ξw F wp ξ w P wp, recall ξ u w = 1 cs,w w cs,u u cs,uξ x x w. (8) u<x w Change variables u = w F,w = w P, x = w F. Then we have ξ w F wp = Note that c w F s,w F Z P\F = z[f] z[f] z[p] = 1 z[f]. 1 1 z[p\f] ( 1) F \F 1 ξ w F wp. (9) F F ( 1) F \F 1 Z P\F F F

46 Remark Stembridge (2001) classified dominant λ-minuscule element of Weyl groups and found another two families other than 15-classes of Proctor s., which are the cases of non-simply laced Dynkin diagrams. All our arguments can be applied to these cases. Replace colored d-complete posets by a heap H(w) of a dominant λ-minuscule element w. Definitions of E P (F) and B(D) should be slightly modified. Then the same ED-hook type formula of the theorem holds.

47 Introduction Main Theorem Idea of Proof Outline Equivariant K -theory Kashiwara thick flag variety G/P Equivariant K -theory K T (G/P ) and localization map Littlewood-Richardson coefficients Chevalley formula Application to d-complete posets Setup Billey type formula Chevalley formula for d-complete poset Example Birds

48 Smallest Bird Figure: e 6 [1; 2, 2]

49 1 1 Example z[1] = z 1 z2 2 z2 3 z 4z5 2 z 6, z[2] = z 1 z 2 z 3 z 4 z 5 z 6, z[3] = z 1 z 2 z 3 z 4 z 5, z[4] = z 3 z 4, z[5] = z 1 z 2 z 3 z 5 z 6, z[6] = z 5 z 6, z[7] = z 1 z 2 z 3 z 5, z[8] = z 3, z[9] = z 5, z[10] = z [P] = (1 z[i]) i=1 There are 29 order filters. F 7 = {1, 2, 3, 5} E P (F 7 ) = {{1, 2, 3, 5}, {1, 2, 5, 8}, {1, 2, 3, 9}, {1, 2, 8, 9}, {1, 7, 8, 9}}, (1 z[1])(1 z[2])(1 z[3])(1 z[5]) G P/F7 = + (1 z[1])(1 z[2])(1 z[5])(1 z[8])z[3] + (P) (P) (1 z[1])(1 z[2])(1 z[3])(1 z[9])z[5] + (1 z[1])(1 z[2])(1 z[8])(1 z[9])z[3]z[5] + (P) (P) (1 z[1])(1 z[7])(1 z[8])(1 z[9])z[2] (P) (1 z[1])f(z) = (P) where F(z) = 1 z 1 z 2 z3 2 z 4z 5 z 1 z 2 z3 2 z 4z 5 z 6 z 1 z 2 z 3 z5 2 z 6 z 1 z 2 z 3 z 4 z5 2 z 6 + z 1 z 2 z3 2 z 4z5 2 z 6 z 2 1 z2 2 z2 3 z 4z 2 5 z 6 + z 2 1 z2 2 z3 3 z 4z 2 5 z 6 + z 2 1 z2 2 z3 3 z2 4 z2 5 z 6 + z 2 1 z2 2 z2 3 z 4z 3 5 z 6 + z 2 1 z2 2 z2 3 z 4z 3 5 z2 6 z3 1 z3 2 z4 3 z2 4 z4 5 z2 6.

50 B 3 example B 3 Dynkin diagram s 0 = s 1 s 2 w = s 0 (s 1 s 0 )(s 2 s 1 s 0 ) λ 0 -minuscule s 0 s 1 s 2 β 1 β 2 β 3 γ 1 γ 2 γ 3 s 0 s 1 s 0 β 4 β 5 β 6 γ 4 γ 5 γ 6 β 1 = α 0 + α 1 + α 2, β 2 = 2α 0 + 2α 1 + α 2, β 3 = 2α 0 + α 1 + α 2 β 4 = α 0 + α 1, β 5 = 2α 0 + α 1, β 6 = α 0, γ 1 = α 0, γ 2 = α 0 + α 1, γ 3 = α 0 + α 1 + α 2, γ 4 = α 0 + 2α 1, γ 5 = α 0 + 2α 1 + α 2, γ 6 = α 0 + 2α 1 + 2α 2

51 λ = 321, µ = 21 W 1 = 1 (1 z 0 )(1 z 2 0 z 1)(1 z 2 0 z 1z 2 ) z 0 z 1 W 2 = (1 z 0 z 1 )(1 z 0 2 z 1)(1 z 0 2z 1z 2 ) z 0 2 W 3 = z2 1 z 2 (1 z 0 z 1 )(1 z 0 2 z 1z 2 )(1 z 0 2z2 1 z 2) z 0 z 1 z 2 W 4 = (1 z 0 z 1 z 2 )(1 z 0 2 z 1z 2 )(1 z 0 2z2 1 z 2) ED-sum 1 W 1 + W 2 + W 3 + W 4 = (1 z 0 )(1 z 0 z 1 )(1 z 0 z 1 z 2 )

52 Thank you!

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