Kostka numbers Kλμ are non-negative integers indexed by two partitions λ and μ.They equal to the number of semistandard tableaux of shape入andweightμ. Also they equal to the number of K-hives with boundary edge label determined by λ and μ.Their basic property is Kλμ=Kλσ(μ) for any element a of the symmetric group. This is proved by constructing a bijection called the Bender-Knuth transformation between semistandard tableaux. In this paper, we give a perspective of the Bender-Knuth transformation through the hive model

NARISAWA, SHOTA

SHIRAYANAGI, KIYOSHI

English

Kyoto University Research Information Repository

Title Bender-Knuth transformation from a perspective of hives(Computer Algebra --Theory and its Applications)Author(s)NARISAWA, SHOTA; SHIRAYANAGI, KIYOSHICitation数理解析研究所講究録 = RIMS Kokyuroku (2019), 2138:138-146Issue Date2019-12URL http://hdl.handle.net/2433/254896RightType Departmental Bulletin PaperTextversionpublisherKyoto University138Bender-Knuth transformation from a perspective of hives 成澤翔大SHOTA NARISAWA 東邦大学大学院理学研究科GRADIATE SCHOOL OF SCIENCE, TOHO UNUVERSITY * 白柳潔KIYOSHI SHIRAYANAGI 東邦大学理学部FACULTY OF SCIENCE TOHO UNIVERSITY t Abstract Kostka numbers K沖 arenon-negative integers indexed by two partitions入andμ.They equal to the number of semistandard tableaux of shape入andweightμ. Also they equal to the number of K-hives with boundary edge label determined by入andμ.Their basic property is K入"=K畑(μ)for any element a of the symmetric group. This is proved by constructing a bijection called the Bender-Knuth transformation between semistandard tableaux. In this paper, we give a perspective of the Bender-Knuth transformation through the hive model. 1 Introduction Kostka numbers are important and classical numbers in combinatorics and representation theory. Let 入bea partition of n E N andμbe a composition of n which is a partition not assuming the decreasing order. In combinatorics, Kostka number K入μequalsto the number of semistandard tableaux of shape 入andweightμ. Also, when one expresses the Schur function s入asa linear combination of monomial symmetric function m Kostka numbers appear as the coefficients so that s入=~μ, k入μ叫， where入isa partition of n and the sum is over al partitionsμ. In representation theory, Kostka numbers appear as a multiplicity of irreducible representations. More precisely, let S入beSpecht modules which are irreducible modules of the symmetric group and M入bethe permutation module. Then we have Mµ= 〶豆心入µ外A fundamental property of Kostka number is that K>,μ= K知(μ)holds for any element u of the symmetric group. In fact, this is shown by constructing a bijection between semistandard tableaux. The bijection is called the Bender-Knuth transformation [2] which was also used to prove the Littlewood-Richardson rule [12]. *651801 ln@st.toho.u-ac.jp 1kiyoshi.shirayanagi@is.sci.toho-u.ac.jp 139On the other hand, the hive model was introduced by A.Knutoson and T.Tao [9]. It is a triangular graph like (2). If a hive's boundary edge label is determined and the hive satisfies some conditions, then it is called K-hive. And we see that a K-hive 1-1 corresponds to a semistandard tableau. Hence, Kostka number equals to the number of some K-hives. In this paper, we give a perspective of the Bender-Knuth transformation from the hive model, namely, we construct a bijection between K-hives corresponding to the Bender-Knuth transformation. 2 Definitions 2.1 Kostka number and related notation In this section, we will define Kostka numbers and explain the Bender-Knuth transformation. Through-out this paper, 6n is the symmetric group of degree n . Definition 1 入＝ （入1,.. , 入k)is called a partition of n E N if it satisfies入iE Z;,o(i = 1, ... ,k), 入1+・・・＋入k=n and >.1 2・ ・ ・2入k.Also, each入iis called a pa廿. We denote the sum of al parts by I入I-If入satisfiesconditions入iE Z;,o andふ＋・・・十入k= n (but not necessarilyふミ・・・ミふ） then it is called a composition. We define an action of 6n on compositions as follows. Letμ= (μ1, .. , μk) be a composition of n. Then, forび E6n, u・ 入：＝（入r;(l),.. , 入r;(k)-Definition 2 Let入＝（ふ，．．．，ふ） be a partition of n. The Young diagram of shape入isan array of n boxes having k left-justified rows with row i containing入iboxes for 1 :;i :;k. Example 1 Ifn = 3, partitions of3 are (3), (2, 1), (1, 1, 1). Then the corresponding Young diagram are respectively ITIJ EP LJ・A Young diagram can be filed with a number per box. Definition 3 Let入＝（入1,...'入り bea partition of n andμ= (μ1, .. , μs) be a composition of n. T is a semistandar-d tableau of shape入andweightμif it is an array obtained by丘llingin the boxes with positive integers such that the number of i's'in Tisμ,, its rows weakly increase and its columns strictly increase. Definition 4 Let入＝（入1,...'入k)be a partition of n andμ= (μ1, .. , μs) be a composition of n. Kostka number-K入μis de且nedas the number of semistandard tableaux of shape入andweightμ, namely k入μ={ Semistandard tablueaux of shape入andweightμ} 140Example 2 Let入=(3, 2) andμ= (2, 1, 2). Then Semistandard tableallJ(of shape入andweightμare 凸凸Then Kost細 numbersK入μ=K(3,2),(2,l,2) = 2. Also, we take another weightμ'= (2, 2, 1) (= (2 3)・μ),that is we consider the samistandard tableaux of shape入=(3, 2) and weightμ'= (2, 2, 1). Then they are 屈州ThusK研=K(3,2),(2,2,l) = 2. This example is not a special case. The following proposition shows this property of Kostka numbers. Proposition 5 ([2], [11]) Let入bea partition of n andμbe a composition of n. Then for al a E 6n we have K入μ=K板(μ)・Proof We will construct a bijection between semistandard tableaux of shape入andweight Jl and semistandard tableaux of shape入andweight a(μ). It suffices to show the proposition in the case び =(ii+l).Take a semistandard tableau T of shape入andweightμ. First, we fix blocks which are pairs i and i + 1 in each column. We call the pairs fixed and al other occurrences of i or i + 1 free. In each row, switch the number of free i's and i + l's, more precisely if the row consists of k free i's followed by l free i + l's then replace them by l free i's followed by k free i + l's. Clearly, this is a bijection and the map yields a semistandard tableau. The map is called the the Bender-Knuth transformation. 1 Example 3 We give an example of the Bender-Knuth transformation. Suppose入=(4,3), μ= (1,3,2,1) and び=(1, 2). Then a(μ) = (3, 1, 2, 1). The left of (1) is a semistandard tableau of shape入andweightμ, and the right of (1) is a semistandard tableau corresponding to the left of (1) by the the Bender-Knuth transformation. We can see that its shape is入andits weight is a(μ). 帽屈 (1) Recall weightμcontrols the multiplicity of entries. We can check switching 1 and 2. 2.2 Hive Now we introduce the hive model, see [1] and [7]. Definition 6 In the ve廿exrepresentation, An n-hive graph is a labelling of an vertices of equilateral triangular graph. 141The following is the example of 4-hive graph in the vertex representation. a03 ／＼ ao2 a13 / " / " ao1 a12 a23 / " / " / " aoo au a22 a33 A hive graph h邸 threedifferent type rhombus (3) which are called the elementa内/rhombus. a ／＼ a — b b C b― a "-I "- ＼／ /¥/ C― d d d― C left-learning upright right-learning Also, for the elementary rhombus (3) the rhombus inequality takes the following form (4). b+c 2'.a+ d.(2) (3) (4) A hive graph has other expressions which are very useful. The edge rep冗 sentationis a labelling of al edges of a hive graph satisfying triangular conditions 1 = a+ /3for elementary triangles (6) and betweenness condition (7). If n = 4, hive graph in the edge representation is below Elementary triangles is below. Also, betweenness conditions are: ＾ 匈 3 813 K "/13→0<02 /312 0<13 823 K 712 丑 123→匈 1 f3u 0<12 /32 0<23 /33 ← ;ll V ;22 V ;33→ ／ ← 1→ a (3 /3 a ← 1→ ／ヽ(5) (6) °'i-1,j-1 :> 年j:> °'i-1,j, /3ij :> /3i,j-1 :> /3i+1,j, "/i,j-1 :> "/ij :> "/i+l,j (7) We can change the vertex representation to the edge representation in the following manner. In each edge between neighbouring vertices labelled a and b, edge label determined by means of the difference b -a ifbis on the right of a. In the illustration of the edge representation (5), parameters°'ij, /3ij, "/ij is defined by°'ii = aij -ai j -1, 九=aij -ai-1か冗=aij -ai-11-1-In the edge representation, the rhombus inequality takes the form of a::>:'Y, f3 :> ,5 in the labelling of below (8). °' ~ ~ ~ (8) 142To introduce another expression of the hive graph, we introduce some notations. For three different elementary rhombi (9) 初/3i,J-~ 三+l,J布+l,J :~~: 宵j°'i-1,j-~ ビ万'1布，j-1(9) we define parameters Lij, Uij and R;j by L;i = /3i,j-1 -/3;+1,i ='Yii―"fi+l,J, Uii =°'ii -°'i-1,i =恥ー/3i,j-1,Rii =伍-1,j-1-°'ij ='Yi,j-1 -'Yij・ (10) (11) (12) They are called the gradient of the corresponding left-leaning, upright and right-leaning rhombi, respec-tively. Note that we can see L;j =江―:~U;,k 一こい Ui+l,k ・A gradient representation is a labelling of boundary edges and gradients of giving the gradients of one or other of its three sets of right-leaning, upright or left-leaning elementary rhombi. a3 f31 a3 f31 °'2 /32 °'2 /32 L12 R13 a, /3 L23 £13 R12 R23 "/1 "/2 "/3 "/1 "(2 "/3 "/1 ,2 "/3 (13) In a gradient representation, the rhombus inequalities change to Lij 2: 0, U匂 2:0, 脱 2:0, respectively. Now we define K-hive which is a hive graph satisfying some condition. As will be seen, it 1-1 corre-sponds to a semistandard tableau. If vertexes of a hive graph are integers we call it integer-hive gmph. Definition 7 Let入bea partition with l(入)<:'. n andμbe a composition with l(μ) <:'. n, also I入I= lμI. In the vertex representation, a K-hive is an integer hive graph satisfying the rhombus inequality (4) for left-leaning and upright (but not right-learning) with boundary edge labels determined by a。,=0 (i=O, .. ,n), ai, =μ1 + .. +μ, (i = 1, .. ,n), a,n =入1+ .. 十入, (i = 1, .. , n). We denote by 1-l(入,μ) the set of K-hives with boundary edge labels determined by入andμ.If n = 4,K-hives take the right form of (14) also in the edge representation it takes the left form of (14). ?゜ ?? ?? |入I=lμI μ1 μ1 +μ2 μ1 μ2 μ3 (14) 143Example 4 If入=(2, 2, 1) andμ= (2, 1, 2), the left of (15) is a example of K-hive with boundary edge label determined by入andμ.Also, in the edge representation, we have the right of (15). ??????????????????????ヘ0 2 ペー 2~0 2 0 2 ペー 2 天 2~0 2 0 1 1 1 ← 2'-./ 1'-./ 2→ (15) 3 Relationship between Kostka numbers and hives In this section, we show that the number of semistandard tableaux of shape入andweightμequal to the number of K-hive with boundary edge label determined by入andμ.Also, we explain the gradient representations in detail. We will start with the following Lemma. Lemma 8 如＝入iー と囚 =/1,i一 Lu如i<k i>k Proof We take a K-hive H and consider the semistandard tableau T corresponding to H. Since T ・1s sem1standard entries o f ith row of T are grater than i. Then 入，=~i:<,k Uik. Thus we get 如＝入,-~i<k Uik• Also Since T is semistandard again, the entries i isin from the top to the ith row, namelyぃ＝区陀iUki・Then we get Uii =μi一区k>iUki・ I Proposition 9 Let入bea partition andμbe a composition of n with l (入），l(μ)<: n. Then we have K入μ=#1i(入，μ).Proof To prove it, we construct a bijection between semistandard tableaux of shape入andweight μand K-hives with boundary edge labels determined by入andμ.We define a map from tableau T to a hive graph H by U;j = the number of j in ith row (16) and adding boundary edge label入andμ.Also, the inverse map is constructed in the following manner. 入andμdetermineshape and weight respectively. We can find entries of ith row as follows. From i + 1 ton can be seen directly from Uij (i < j~n). Since Tis semistandard, the entries of ith row is greater than i. Thus, it suffices to find the number of i, this is Lemma 8. Then, we arrange the entries in the weakly incre邸 ingorder from left to right. Then, we have only to show that the map from T to H is a map from semistandard tableaux to K-hives and its inverse is a map from K-hives to semistandard tableaux. Recall that U,j implies the number of j in ith row and J-1 Lij = Lui,k一 Lui+l,kk=l k=l = (# of entries~j -1 in row i) -(# of entriesさjin row i + 1). 144Take T and the image H of the map, we show H is a K-hive, namely H satisfies rhombus inequality for left-learning and upright. By definition, U;j 2 0. Since T is semistandard, Lり 20. Conversely, take K-hive H and the image T of the inverse map. Since Uり20, each row of T contains non-negative numbers of each distinct entry. Also since Lij 2 0, the number of entries in ith row above j's in i + 1th row is less than j. Then T is semistandard. 4 Bender-Knuth transformation from a perspective of hives Now we give a perspective of the Bender-Knuth transformation from the hive model, namely, we give another proof of Proposition 5. Concretely, our aim is to construct a bijection between hives. Consider the situation such邸：Proposition 9 叶一? ?Proposition 51 I Our aimProposition 9 叶一2 1 2 (17) In the Bender-Knuth transformation, it is constructed by swapping the number of entries in each row aftcr fixing somc blocks. To rcproducc this bijcction, wc will start by idcntifying thc numbcr of fixcd blocks from a K-hive. Take u = (ss + 1)E <5n and we consider the ith and i + 1th rows which have fixed blocks. Now possible situations for positions of s and s + l are: 三 Iロニ E] I I 1s+/ I I I We call them type 1, type 2, type 3 and type 4 from left to right. The number of boxes up to a first fixed box is江―:,i仰＝江ョ如1,kin type 1 and type 2 and江―:,iU;k in type 3 and type 4, respectively. On the other hand, the number of boxes up to a last fixed box is LいU;k=Lにiui+l,k in type 1 and type 3 and区:'.iU; 十l,kin type 2 and type 4, respectively. Now we denote by FJi+1 the number of fixed boxes between the ith row and i + 1th row. If there is a fixed boxes between the ith row and i + 1th row, then FJi+l can be expressed byとにiu, 十1,k-江―:,iUik・Otherwise, FJi+l is defined as 0. 145Definition 10 Setび=(s, s + 1)E 6n. Let L;j, U;j and R;j be gradients of a H(入，μ)E如）（入，μ)and Lら， U[1and Rらbegradients of a H'(入，μ')E Ji(n)(入，μ).Then the map¢: 砂（入，μ)→印（い(μ))is defined as follows: µ':=び •µandヽ~ー??、ヽ~??― ― ― ― ?????????――??）ー????????，??? ??，????ヽ~??＿?? ???― ― ??? (18) Our main result is as follows. Theorem 11 c/>: 11,(n)(入，μ)→砂）い(μ))is an involution. Proof Clearly, c/> is an involution, so we will check only the well-definedness, namely H(入，μ)is just a K-hive. By definition,μ'= u(μ). Then it suffices to show that H'(入，μ)satisfies the rhombus inequality for upright and left-le皿 ing.We have s+l s-1 Uし=U;,s+l + F;,i+l―窮―1,i= ui,s+1 + F;•i+l -(区U;,k 区ーU;-1,k) (19) k=l k=l s-1 s = Fi••+1 + L U;-1,k -L い=F;•i+l + Li-1,s 2': 0 (20) k=l k=l Similarly, we can get U;,s+l 2': 0. Also J-1 J Lij =区U臼ー 区UI+1,kk=l k=l (21) J-1 = L い +uし+ULs+l -(t ui+l,k + U:+1,s + U:+1,s+1) (22) kfcs,s+l kfcs,s+l J-1 J =L叫—区Uf+1,k 戸 0 (23) k=l k=l Thus H'(入，μ')is a K-hive Here let us calculate an involution¢in detail. Example 5 Suppose that入=(3,2,0), μ= (2,2, 1) and a= (23) E釣.Take the left of (24), then the corresponding K-hive is the right of (24). 0M2 oA_1い? ? ? ?? ? ? ?(24) where FJ2 = 0, F;3 = 0, U伍=U13 = 0, U{3 = U12 = 1, U匂=U22 =入2-U23 = 1. We can see that it agrees with (17). 1465 Conclusion We gave a perspective of the Bender-Knuth transformation through the hive model (11). More precisely, we constructed a bijection between K-hives. We have dealt with K-hives which are the hive models corresponding to semistandard tableaux. On the other hand, the hive is related not only Kostka numbers but also Littlewood-Richardson coefficients. It is called LR-hive. There is a relationship between Kostka numbers and Littlewood-Richardson coefficients. If the relationship is described using the hive model, we may get a good understanding of the relationship and the hive model. References [1] Olga Azenhas, Ronald C King, and Itaru Terada. The involutive nature of the littlewood-richardson commutativity bijection. ar Xiv preprint ar Xiv: 1603. 05037, 2016. [2] Edward A. Bender and Donald E. Knuth. Enumeration of plane partitions. J. Combinatorial Theory Ser. A, 13:40-54, 1972. [3] Anders Skovsted Buch. The saturation conjecture (after A. Knutson and T. Tao). Enseign. Math. (2), 46(1-2):43-60, 2000. With an appendix by William Fulton. [4] Patrick Doolan and Sangjib Kim. The Littlewood-Richardson rule and Gelfand-Tsetlin patterns. 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Proof of the saturation conjecture. J. Amer. Math. Soc., 12(4):1055-1090, 1999. [10] I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford Classic Texts in the Physical Sciences. The Clarendon Press, Oxford University Press, New York, second edition, 2015. With contribution by A. V. Zelevinsky and a foreword by Richard Stanley, Reprint of the 2008 paperback edition [ MR1354144]. [1] Bruce E. Sagan. The symmetric group, volume 203 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2001. Representations, combinatorial algorithms, and symmetric functions. [12] John R. Stembridge. A concise proof of the Littlewood-Richardson rule. Electron. J. Combin., 9(1):Note 5, 4, 2002. 

Bender-Knuth transformation from a perspective of hives (Computer Algebra --Theory and its Applications)

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Bender-Knuth transformation from a perspective of hives (Computer Algebra --Theory and its Applications)

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