Sergey Fomin* and. Minneapolis, MN We consider the partial order on partitions of integers dened by removal of
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1 Rim Hook Lattices Sergey Fomin* Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA Theory of Algorithms Laboratory St. Petersburg Institute of Informatics Russian Academy of Sciences and Dennis Stanton** School of Mathematics University of Minnesota Minneapolis, MN Abstract We consider the partial order on partitions of integers dened by removal of rim hooks of a given length. The isomorphism between this poset and a product of Young's lattices leads to rim hook versions of Schensted correspondences. Analogous results are given for shifted shapes. 1. Main Results A shape, or Young diagram, is a nite order ideal of the lattice P 2 = f(k; l) : k; l 1g. Each shape = f(i; j) : 1 j i g can be identied with the corresponding partition = ( 1 2 : : : ). Shapes form the Young lattice Y (see, e.g., [St86]). In drawing the shapes, we use the so-called \English notation;" for example, the partition (5; 3; 1; 1) is represented by the shape We denote by # the number of boxes in a shape. The i'th diagonal of a shape is the set f(k; l) : l? k = ig. A rim hook is a contiguous strip of boxes in P 2, which has at most one box on each diagonal. Throughout the paper a positive integer r is xed, and all the rim hooks contain exactly r boxes, with the exception of Denition 3.4(3). *Partially supported by the Mittag-Leer Institute. **Partially supported by the Mittag-Leer Institute and by NSF grant #DMS Mathematics Subject Classication. 05A. Key words and phrases. Young lattice, rim hooks, dierential posets. : Typeset by AMS-T E
2 2 1.1 Denition. Let and be shapes such that? is a rim hook. Then we write m. We write if there exists a sequence of shapes = 0 m 1 m m k =. In other words, means that can be obtained by deleting rim hooks from. If, then is said to be r-decomposable. The poset of all r-decomposable shapes ordered by is denoted by RH r and called the rim hook lattice. (We shall prove that this is indeed a lattice.) The following result is essentially known; however, we could not nd it elsewhere stated in this explicit form. 1.2 Theorem. The rim hook lattice RH r is isomorphic to the cartesian product of r copies of the Young lattice: RH r = Y r : The proof of this theorem is given in Sec.2. Figures 1 and 2 show the lattices RH 2 and RH 3, respectively; on the latter the underlying poset 3P 2 is highlighted. 1.3 Denition. Let P be a graded poset, K a eld of zero characteristic, and KP a vector space with a basis P. Dene the up and down operators U; D 2 End(KP ) by Ux = Dy = y covers x y covers x 1.4 Proposition. [Fo4,St88] The up and down operators in the Young lattice Y satisfy DU? UD = I where I is the identity transformation. Thus Y is a self-dual graph [Fo1] or a dierential poset [St88]. Generally, two graded graphs G 1 and G 2 with a common set of vertices and a common rank function are called r-dual [Fo1,Fo3] if the up operator U 1 in G 1 and the down operator D 2 in G 2 satisfy y ; x : D 2 U 1? U 1 D 2 = ri : It is easy to see [Fo1, Lemma 2.2.3] that if the graphs G 1 and G 2 are r-dual, and the graphs H 1 and H 2 are s-dual, then G 1 H 1 and G 2 H 2 are (r + s)-dual. 1.5 Corollary [SS90, Sec. 9, (9)]. The up and down operators in the rim hook lattice RH r satisfy the relation DU? UD = ri. Thus RH r is an r-self-dual graph, or r-dierential poset. One can therefore apply to RH r any of the enumerative results about such graphs given in [St88, Fo1,
3 3 Ro91, etc.]. Moreover, this will allow us to construct an analogue of the Schensted algorithm for the rim hook lattice, following [Fo2, Fo3]. This algorithm establishes a bijection between pairs of paths in RH r (\standard rim hook tableaux" [SW85]) and r-colored permutations. It is immediate from Theorem 1.2 that this algorithm can be viewed as a \direct product" of r copies of the standard Schensted algorithm (see, e.g., [Sa90]) running in parallel. The resulting algorithm coincides with one of [SW85], where it was originally described in terms of insertion steps. The next two corollaries provide examples of applications of general enumerative theory of r-dierential posets to the rim hook lattice. Let e r () denote the number of saturated chains = 0 m 1 m m n = (i.e., the number of standard rim hook tableaux of shape ). Similarly, we let e r (=) denote the number of \standard skew rim hook tableaux of shape =," i.e., the number of paths = 0 m 1 m m n = in the rim hook lattice. 1.6 Corollary. For 2 RH r, (i) (ii) #=rn #=rn e 2 r () = rn n! ; e r () = #fr-colored involutions in the symmetric group S n g where \r-colored involution" means a symmetric n n-matrix containing exactly one nonzero entry in each row and column; this entry should be one of 1; 2; : : : ; r. Proof. See, e.g., [Fo1, (1.5.19)] and [Fo2, Corollary 3.9.4]. The following result is an analogue of the formulae of [SS90]. 1.7 Corollary. Let ; 2 RH r, # = rk, # = r(k + n? m) (n and m are xed). Then #=r(k+n)e r (=) e r (=) = j r j m j Proof. See [Fo2, Corollary 3.8.3(iv)]. n j j! #=r(k?m+j) e r (=) e r (=) : In Sec. 3 we give the analogues of the above results for the shifted shapes. Acknowledgements. We are grateful to Anders Bjorner who asked whether the approach of [Fo1, Fo2] can be applied to rim hook tableaux. We thank Curtis Greene and Richard Stanley for helpful comments. Figures were produced by Curtis Greene using the Mathematica TM package [GH90]. This paper originally appeared in March 1992 as Report No. 23 of Institut Mittag-Leer.
4 4 2. Fairy Sequences 2.1 Denition. A map f : Z?! Z is called a fairy sequence if the following conditions hold: (i) f(i) f(j) whenever 0 i j ; (ii) f(j) f(i) whenever j i 0 ; (iii) jf(i)? f(i? 1)j 1 for all i ; (iv) f(i) = 0 if jij is suciently large: The following lemmas concerning fairy sequences are fairly straightforward. 2.2 Lemma. There is a bijective correspondence between shapes (Young diagrams) and fairy sequences. This bijection (denoted 7! f ) is given by f (i) = #f boxes on the i'th diagonal of g : 2.3 Lemma. A shape is r-decomposable if and only if the corresponding fairy sequence f = f satises the following condition: (1) ia( mod r) f(i) = ib( mod r) f(i) for any a; b 2 Z: 2.4 Denition. Let f (0), : : :,f (r?1) be fairy sequences. Then will denote the sequence dened by (2) f(i) = f = hf (0) ; : : : ; f (r?1) i r?1 i + k f (k) r k=0 where [: : : ] stands for the integer part. 2.5 Lemma. We have (3) f(i)? f(i? 1) = f (k) i + k r where k = (?i) mod r.? f (k) i + k r? Lemma. For any fairy sequences f (0), : : :,f (r?1), the sequence is fairy and r-decomposable. f = hf (0) ; : : : ; f (r?1) i Proof. The rst property follows from Lemma 2.5, the second one from ia( mod r) f(i) = k j f (k) (j) :
5 5 2.7 Lemma. Any fairy and r-decomposable sequence f can be uniquely represented as f = hf (0) ; : : : ; f (r?1) i, where f (0), : : :,f (r?1) are some fairy sequences. Proof. Given such a sequence f, one can inductively use (2) to nd the values f (k) (j), starting with f (k) (j) = 0 for j 0. The equality (3) guarantees that the resulting sequences satisfy Denition 2.1,(i)-(iii). The condition (1) for b = a? 1 gives, together with (3), the equality and the proof follows. lim j!1 f (k) (j)? lim j!?1 f (k) (j) = 0 ; Proof of Theorem 1.2. We will prove that the bijection (4) f! (f (0) ; : : : ; f (r?1) ) induces an isomorphism between RH r and Y r. According to (2), the following are equivalent: (i) adding 1 to some f (k) (j) ; (ii) adding 1 to each of f(i); f(i + 1); : : : ; f(i + r? 1) for some i. In view of Lemmas , the operations (i) and (ii) either both preserve the \fairyness" or both don't. To complete the proof, note that (i) corresponds to adding a box to the respective shape, and (ii) to adding a rim hook. Theorem 1.2 can be also proved by means of the approach of [K81]. 3. Shifted shapes In this section we dene the shifted rim hooks (Denition 3.4) and nd the analogous isomorphism theorem (Theorem 3.7) for shifted shapes. 3.1 Denition. Let SemiPascal be the set f(k; l) 2 Z 2 : l > k 1g ordered by inclusion. The nite order ideals in SemiPascal are called shifted shapes; the corresponding distributive lattice is denoted SY. For any 2 SY, let () denote the \symmetrized" shape being a union of and a ipped shifted shape 0 ; formally, 0 = f(l; k? 1) : (k; l) 2 g ; () = [ 0 : Informally, this means that we fold the shifted shape about a diagonal line just to the left of 's main diagonal, and add it to.
6 6 For example, if = (4; 1), then () = (5; 3; 1; 1): : Similarly, two fairy sequences f and g are said to be folded to each other if f(k) = g(1? k) for all k 2 Z. This means that their respective shapes are related by folding across the diagonal in the denition of (). We call f self-folded if f(k) = f(1? k) for all k 2 Z. The next lemma follows immediately from the denition of (). 3.2 Lemma. For any shifted shape, the fairy sequence f = f () is self-folded; namely, (5) f(k) = f(1? k) for k 2 Z : Conversely, any fairy sequence satisfying (5) has a unique representation of the form f = f () for an appropriate shifted shape. It is clear that self-conjugate (left-justied) shapes have fairy sequences f which are symmetric, f(k) = f(?k), for all k 2 Z. We need these fairy sequences for the next lemma, which applies (4) to self-folded shapes (). 3.3 Lemma (cf. [Ol87,MY86,GKS90]). The bijection (4), when restricted to the shifted shape case, reduces to a bijection between (i) r-decomposable self-folded fairy sequences and (ii) r-tuples (f (0) ; : : : ; f (r?1) ) of fairy sequences where f (0) is self-folded, f (i) is folded to f (r?i) for 1 i r?1 2, and if r is even then, in addition, f (r=2) is symmetric. We next dene shifted rim hooks. This denition will allow us to dene the shifted rim hook lattice, SRH r, on all r-decomposable shifted shapes. 3.4 Denition. A shifted rim hook is a convex subset of SemiPascal which satises one of the three following conditions: (1) for some i r, h has exactly one box on each of the diagonals i? r + 1; : : : ; i; (2) for some i; r=2 < i < r, h has two boxes on each of the diagonals 1; : : : ; r? i and one box on the diagonals r? i + 1; : : : ; i; (3) for i = r=2, h has one box on each of the diagonals 1; : : : ; r=2. (In the cases (1) and (2), a shifted rim hook contains r boxes, while in the case (3), r is even, and the shifted rim hook has r=2 boxes.)
7 7 The following picture describes some shifted rim hooks which one can add to the shifted shape 41: M L K H I G F E D C B A For r = 4: BCDE, EF G, K; for r = 5: ABCDE, DEF G, EF GK, GHIK. Shifted rim hooks dene the covering relation in the shifted rim hook lattice SRH r. The posets (lattices) SYand SRH 2 are given in Figures 3 and 4, respectively; the join-irreducible elements are highlighted. The rank function rank r on SRH r is standard for r odd: rank r () = #=r, but this does not hold for r even. For example, for r = 2, rank 2 () is the number of boxes of lying on odd diagonals; rank 2 (542) = 6. The next two lemmas are straightforward corollaries of the denitions. 3.5 Lemma. The bijection $ f () is a poset isomorphism between SYand the coordinate-wise partial order on self-folded fairy sequences. 3.6 Lemma. The coordinate-wise partial order on the set of symmetric fairy sequences is isomorphic to SY. We thus obtain the following result. 3.7 Theorem. If r is odd, then SRH r = SY Y r?1 2 : If r is even, then SRH r = SY SY Y r?2 2 : 3.8 Lemma [Fo2]. Let SY be the graph obtained from SY by doubling the edges which correspond to adding boxes lying outside the rst diagonal. Then SYand SY are dual. Let us now dene the graphs SRHr as follows: take the shifted rim hook lattice SRH r and double its edges which correspond to adding rim hooks of type (1) (see Denition 3.4) with i = rj, j 2, and, in case r is even, those with i = r(j + 2 1), j 1, as well. 3.9 Corollary. The graphs SRH r and SRH r are r+2 2 -dual. In addition, if r is odd, then SRHr = SY Y r?1 2 ; if r is even, then SRH r = SY SY Y r?2 2 : Once a dual graph is constructed, the corresponding Schensted algorithm arises, r?1 as shown in [Fo2, Fo3]. In this case it decomposes into 2 independently running
8 8 ordinary Schensteds, along with one or two (for r odd and even, respectively) independently running \shifted Schensteds" (see [Wo84,Sa87,Ha89,Fo2]). Now we can apply to the shifted rim hook lattices all of the results concerning arbitrary dual graphs (see [St88, St90, Fo1, Fo2]). In particular, we get the following analogue of the Young-Frobenius identity. To state it, let us dene the numbers d r () as follows. Take any decomposition of a shifted shape into shifted rim hooks. Let d r () be the number of shifted rim hooks whose value of i in Denition 3.4 is r(j + 1) or r(j ) for j Corollary. Let e r () denote the number of shifted rim hook tableaux of shape. Then r + 2 n e 2 r() 2 dr() = n! : 2 2SRH r rank()=n To provide an example, let us take r = 2 and n = 3. Note that for r = 2, d 2 () is the number of shifted rim hooks whose value of i is not 1 or 2. The elements of rank 3 in SRH 2 are (6), (4,2), (4,1), (3,2), (3,1), and (5), which respectively have 1,1,3,3,1, and 1 shifted rim hook tableaux for r = 2. For example, for (4,1), the shifted rim hook tableaux are ; 2 ; 2 : The shapes (6), (4,2), (4,1), (3,2), (3,1), and (5) have weights 2 d r() equal to 4, 2, 2, 2, 2, and 4, respectively. Then Corollary 3.10 tells that = 48 = 2 3 3!. References [Fo1] S. Fomin, Duality of graded graphs,. Algebr. Combinatorics 3 (1994), [Fo2] S. Fomin, Schensted algorithms for dual graded graphs,. Algebr. Combinatorics 4 (1995), [Fo3] S. Fomin, Dual graphs and Schensted correspondences, in: Proceedings of the Fourth Intern. Conf. on Formal Power Series and Algebraic combinatorics, Montreal, [Fo4] S. Fomin, Generalized Robinson-Schensted-Knuth correspondence,. Sov. Math. 41 (1988), [GKS90] F. Garvan, D. Kim, and D. Stanton, Cranks and t-cores, Invent. Math., 101 (1990), [GH90] C. Greene and E. Hunsicker, A Mathematica package for exploring posets, Haverford College, [Ha89] M. D. Haiman, On mixed insertion, symmetry, and shifted Young tableaux,. Combin. Theory, Ser.A 50 (1989),
9 [K81] G. ames and A. Kerber, The representation theory of the symmetric group, Vol. 16, in: Encyclopedia of Mathematics and its Applications, (G.-C. Rota, Ed.), Addison-Wesley, Reading, Mass., [MY86] A. O. Morris and A. K. Yasseen, Some combinatorial results involving shifted Young diagrams, Math. Proc. Camb. Phil. Soc., 99 (1986), [Ol87]. Olsson, Frobenius symbols for partitions and degrees of spin characters, Math. Scand., 61 (1987), [Ro91] T. W. Roby, Applications and extensions of Fomin's generalization of the Robinson-Schensted correspondence to dierential posets, Ph.D.thesis, M.I.T., [Sa87] B. E. Sagan, Shifted tableaux, Schur Q-functions and a conjecture of R.Stanley,. Combin. Theory, Ser.A 45 (1987), [Sa90] B. E. Sagan, The ubiquitous Young tableaux, in: Invariant theory and Young tableaux, D.Stanton, Ed., Springer-Verlag, 1990, [SS90] B. E. Sagan and R. P. Stanley, Robinson-Schensted algorithms for skew tableaux,. Combin. Theory, Ser.A 55 (1990), [St86] R. P. Stanley, Enumerative combinatorics, vol.i, Wadsworth and Brooks/Cole, Pacic Grove, CA, [St88] R. P. Stanley, Dierential posets,. Amer. Math. Soc. 1 (1988), [SW85] D. W. Stanton and D. E. White, A Schensted algorithm for rim hook tableaux,. Combin. Theory, Ser.A 40 (1985), [Wo84] D. R. Worley, A theory of shifted Young tableaux, Ph.D.thesis, M.I.T.,
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