1. Introduction KEYS ASSOCIATED WITH THE SYMMETRIC GROUP S 4, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX

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1 Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number KEYS ASSOCIATED WITH THE SYMMETRIC GROUP S 4, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX O AZENHAS AND R MAMEDE 2 Abstract: A variant of the dual RSK correspondence [0, 2] gives a bijection between classes of skew-tableaux and tableau-pairs of conjugate shapes The problem of a matrix realization, over a local principal ideal domain with prime p, of the pair T, Kσ with Kσ a key associated with the permutation σ S t, and T a skew-tableau with the same evaluation as Kσ, is addressed If T corresponds by this variant of the dual RSK to the tableau-pair P, Q of conjugate shapes, there exists a matrix realization for T, Kσ, σ S t, only if P = Kσ [2, 4, 5, 6] This necessary condition has also been proved to be sufficient [7], by exhibiting an explicit matrix realization, in the case the frank word σq is a union of row words whose lengths define the conjugate shape of Q Here, we extend the matrix realization given in [7] to any tableau-pair Kσ, Q of conjugate shapes, with σ S 4 This is carried out by stretching the frank words with shape 2,,,2 which are not the union of one row of length four with one of length two, those associated to 4342 Kǫ,, 0,, 0, with ǫ {423, 432, 423, 432}, to row words of length six associated with the key in S 6 Keywords: Biwords, frank words, keys, dual Robinson-Schensted-Knuth correspondence, matrix realization, shuffle of words AMS Subject Classification 2000: Primary 05E0, 68R5, 5A33; Secondary 05A7, 5A23 Introduction A variant of the dual RSK correspondence [0], Appendix A43 defines a bijection between a tableau-pair P, Q, P [t], Q [n], of conjugate shapes and a class of skew-tableaux whose word is congruent with P, and the word u = u t u with u i the column word defined by the places of the letter i in T, is congruent with Q We observe that french notation is used, see Section 2 Given σ S t, t, a key Kσ associated with σ S t [9, 6], is a tableau whose columns are the t reordered left factors of σ with multiplicity Received February 7, 2006 Research partially supported by CMUC/FCT and SFRH/BSAB/55/2005, grant by the Portuguese Foundation of Science and Technology, FCT, at Fakultät für Mathematik, Universität Wien, Austria 2 Research partially supported by CMUC/FCT

2 2 O AZENHAS AND R MAMEDE l t,, l assigned A tableau-pair Kσ, Q, σ S t, of conjugate shapes is in bijective correspondence with the class of skew-tableaux whose word is in the Knuth class of Kσ and u is the frank word in Q whose column lengths are the σ permutation of the column lengths of Q in reverse order Moreover, the frank word u is a union of rows with lengths given by the conjugate shape of Q iff the word of the skew-tableau is in the shuffle of the columns of Kσ The Knuth class of a key Kσ, σ S t, contains the shuffle of their columns In general, unless the permutation σ S t, t 3, or the permutation word σ S t, t 4, satisfies certain conditions, we do not have equality In S 4, equality fails only for the permutations ǫ {423, 432, 423, 432} The words congruent to the key Kǫ, l 4, l 3, l 2, l with l 4, l 2 > 0, either are in the shuffle of the columns or in the shuffle of the columns and 4342 Their associated frank words are therefore either the union of rows or the union of rows and frank words associated with 4342 The frank words associated with 4342 are those of shape 2,,, 2 which can not be written as a union of one row of length four with one of length two However the entries of those frank words satisfy a row condition which allows us to stretch them to a row of six columns associated with the key in S 6 Given a pair P, Q of tableaux of conjugate shapes which corresponds by the variant of the dual RSK to a skew-tableau T, we consider the problem of a matrix realization, over a local principal ideal domain with prime p, of a pair T, F with F a tableau with the same evaluation as T Section 4, Definition 4 The set of tableaux in [t] with evaluation m,, m t is a rooted tree with root the unique row tableau of evaluation m,, m t and where the key of evaluation m,, m t is a leaf [3] We focus our study in the case F is a key associated with σ S t However, in example 4, a matrix realization is given when F is the root of the tree of the tableaux of evaluation m, n, and, in this case, P is running over the tableaux in this tree It has been shown in [5] that there exists a matrix realization for T, Kσ only if P = Kσ Equivalently, only if u is the frank word in Q whose column lengths are in the reverse order of the permutation σ of the column lengths of Q This necessary condition has also been proved to be sufficient [7], by exhibiting an explicit matrix realization, in the case the frank word u in the class of Q is a union of row words whose lengths define the conjugate shape of Q By stretching the frank words of shape 2,,, 2, associated with 4342, to a row word of length six associated with the key

3 KEYS, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX in S 6, we extend the matrix realization given in [7] to T, Kσ with σ S 4 The paper is divided in five sections In section 2, definition and properties of keys and frank words are given and a variant of the dual RSK correspondence for skew-tableaux is considered [0], Appendix A43 In section 3, we study in detail the words congruent with keys associated with S 4 and the frank words with four columns Special attention is addressed to the words congruent with keys Kǫ, ǫ {423, 432, 423, 432} and their associated frank words In section 4, regarding the correspondence between tableau-pairs of conjugate shape and skew-tableaux given by a variant of the dual RSK, the concept of matrix realization, over a local principal ideal domain with prime p, of a pair T, F with T a skew-tableau and F a tableau with the same evaluation as T, is discussed A matrix realization for the pair T, Kσ, with σ S 4, is exhibited, reducing its construction to the case the frank word σq is a union of rows whose lengths define the conjugate shape of Q In the last section, remarks and extensions of this matrix construction, in some special cases, are made for σ S t, t 5 2 Keys, frank words and a variant of the dual RSK correspondence 2 Keys and frank words Let N be the set of positive integers with the usual order Given k, t N, k t, [k, t] denotes the set {k,, t} in N When k =, we put [t] := [, t] We denote by [t] the free monoid in the alphabet [t] and by λ the empty word A partition is a sequence of nonnegative integers a = a, a 2,, all but a finite number of which are nonzero, such that a a 2 The maximum value of i for which a i > 0 is called the length of a If the length of a is zero, we have the null partition a = 0, 0, If a i = 0, for i > k, we write a = a,, a k as well Sometimes it is convenient to use the notation a = a m, am 2 2,, am k k, where a > a 2 > > a k and a m i i, with m i 0, means that a i appears m i times as a part of a Thus, every partition can be written as a = t l t,, 2 l 2, l for some t and nonnegative integers l i, i t Given the sequence l t,, l of nonnegative integers, we associate the partition m = t k= l k,, l t +l t, l t, the conjugate partition of t l t,, 2 l 2, l 2 Let S t denote the symmetric group of degree t We define the action of σ S t on the partition m by putting σm := m,, m t, where m σi = t k=i l k, i =,, t We have t l t,, 2 l 2, l = t i= m σi The reverse sequence

4 4 O AZENHAS AND R MAMEDE of m,, m t is m,, m t rev := rev σm = m t,, m, where rev = t 2 denotes the reverse permutation in S t Given a word w = x x k over the alphabet [t], we denote by w j the multiplicity of the letter j [t] in w Here k is the length of w, denoted by w The sequence w,, w t is called the evaluation of w, denoted by evw The length and evaluation of the empty word are zero The word w, with k, is said a row if x x k, and a column if x > > x k If w is a column, in the planar representation, the letters are displayed in 5 a column by decreasing order from top to bottom For example, 2 is the planar representation of 52 Let V denote the set of all columns in [t] Every word in [t] has a unique factorization as a product of a minimal number of columns w = v v 2 v r, with v i V We call it the column factorization of w and denote it occasionally by v v 2 v r The shape of w is the sequence w = v,, v r of the lengths of the column factors v i of w For instance, w = is the column factorization of w Given the sequence of nonnegative integers u = u,, u r, we define the word um := u u + u 2 u + u r + + u u r + + u + whose shape is the vector obtained by suppressing in u the null entries [6] We identify u with the shape of um For instance, 3,, 2M = The underlying set of a column defines a bijection v {v} between the set V and the family 2 [t] of subsets of [t] According to this bijection we often identify a column with its underlying set This bijection allows to extend to V the order on 2 [t] by letting u v if and only if there is an injection i : {u} {v}, x ix For instance, In particular, if {u} {v} we have u v We define another order on 2 [t], and extend it to V, putting {u} {v} if and only if there is an injection i : {v} {u}, x ix [6] For instance, A word w = v v 2 v r, v i V, is called a tableau if v v 2 v r The shape of a tableau is, therefore, a partition For instance, =

5 KEYS, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX 5 is a tableau of shape 4, 2, 2, = 4, 3 0, 2 2, The conjugate partition 4, 3,, defines the length of the rows of the tableau The plactic or Knuth congruence on words, over the alphabet [t], [2, 3, 4], is obtained by means of Schensted s construction [9] As usual Pw denotes the unique tableau congruent with w [t] and Qw the Q-symbol of w, the recording tableau of the row insertion of w in the Schensted s construction The RS-correspondence w Pw, Qw is summarized as follows: each plactic class contains a unique tableau P; and the elements of the plactic class of a tableau P are in bijection with the set of standard tableaux of the same shape as P A tableau whose columns are pairwise comparable for the inclusion order is called a key [6] That is, a tableau u r u is a key if {u r } {u 2 } {u } Equivalently, a key K is the tableau whose shape is the conjugate of its evaluation by nonincreasing order This means that the key of evaluation m,, m t is the unique tableau such that stdk T evkm, where std stands for standardization and T for transposition For instance, K = = is the key with evaluation 3,,, 0, 4, the unique tableau with evaluation 3,,, 0, 4 such that stdk T Keys are also tableaux whose columns are the left reordered factors of a permutation with multiplicity assigned For each pair consisting of a permutation σ S t, written as a word σ = a a t [t], and a sequence of nonnegative integers l t,, l, Ehresmann [9] associated a key, here denoted by Kσ, l t,, l, putting Kσ, l t,, l := r σ,t l t r σ,t l t r σ, l, where r σ,k is the column with underlying set {a,, a k }, k t This key is the tableau with shape t l t,, 2l 2, l and evaluation σm When σ = id, Kid, l t,, l is said the Yamanouchi tableau of evaluation m, that is, the tableau whose shape is the conjugate of the evaluation The congruent words are called Yamanouchi words of evaluation m A word w [t] is said frank [6], [0], Appendix A5, if its shape is a permutation of the shape of Pw The following theorem, proved by Lascoux

6 6 O AZENHAS AND R MAMEDE and Schützenberger in [6], shows that the frank words, in a plactic class, are in bijection with the set of permutations of the shape of the tableau in that class Frank words are completely determined by the conditions imposed on their Q symbols Theorem 2 Let Q be a tableau with shape m For each permutation σ S t, there exists one and only one word σq Q with shape σm σq is such that the Q-symbol is σmm Keys and frank words are therefore related as follows Corollary 22 The frank words J with shape σm are those whose transposition of the Q-symbol is the standardization of the key K with evaluation σm, that is, QJ = stdk T The frank words J of shape σm rev are those such that QJ = evacqj = stdk T, with K = evac K the key of evaluation σm rev, where evac stands for the evacuation operation A skew-tableau T in [t] [5] is a tableau on the alphabet [t] { }, where the extra letter is such that < < < 2 < < t The word wt of the skew-tableau T is the word in [t] obtained by eliminating from T the extra letter, and the evaluation of T is the evaluation of wt Let a be the partition defined by the number of letters in each column of T Then, if c is the shape of T, c/a, called the skew-shape of T, denotes the sequence of number of letters of wt in each column of T, from left to right In particular, a tableau in [t] is a skew-tableau with a = 0 For example, T = is a skew-tableau of skew-shape 6, 6, 5, 3, 3, 2/4, 4, 2, 2, 2, 0 = 2, 2, 3,,, 2, and its planar representation is A skew-tableau with word w w n and skew-shape w,, w n is in the compact form if the inner shape a = n i=2 w i v i,, w n v n, 0 where v i is the left factor of w i of maximal length satisfying w i v i Using

7 KEYS, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX 7 jeu de taquin [0, 20] in consecutive columns from right to left, every skewtableau can be put in the compact form We identify skew-tableaux having the same compact form The skew-tableau 2 is in the compact form 22 A variant of the dual RSK-correspondence Regarding the matrix problem addressed in the Introduction, we consider a variant of the dual Robinson-Schensted-Knuth correspondence [2], [0], Appendix A43, to establish a bijection between tableau-pairs P, Q of conjugate shapes and skew-tableaux in the compact form u u u Let = k be a biword with no repeated biletters, where v v v k u u,, u k [n] and v,, v k [t] Sorting the biletters of by v nondecreasing rearrangement with respect to the anti-lexicographic order with priority on the first row, we get Σ = f n f n, 3 w w n where evu = w,, w n and w = w w n [t], w i V {λ}; and by u nonincreasing rearrangement of the biletters of for the lexicographic v order with priority on the second row, we get Σ Jt J = t m t m, 4 where evv = J,, J t and J = J t J [n], J i V {λ} Consider the transformation Σ Σ defined by sorting the biletters of Σ in nonincreasing rearrangement with respect to the lexicographic order with priority on the second row, and by sorting the biletters of Σ in nondecreasing rearrangement with respect to the anti-lexicographic order with priority on the first row From Greene s theorem, we have Lemma 23 a The transformation Σ Σ establishes a bijective correspondence between the k-tuples of disjoint nondecreasing subwords of J = J t J and those of decreasing subwords of w = w w n b The tableaux Pw and PJ have conjugate shapes with evw rev = J t,, J and evj = w,, w n

8 8 O AZENHAS AND R MAMEDE u A biword without repeated biletters determines a unique pair of v u J biwords Σ =, Σ w = defined as above we write v u for v v by nonincreasing for u by nondecreasing order Two biwords are said equivalent if they consist of the same biletters We consider the variant of the dual RSK-correspondence, here denoted by RSK, [0], Appendix A43, for an arbitrary biword without repeated billeters by u RSK P, Q, v where P = Pw and Q = PJ This pair of tableaux is related as follows u J Theorem 24 Let Σ = and Σ w = as before v a PJ is the unique tableau of evaluation w,, w n such that Qw = stdpj T b Pw is the unique tableau of evaluation J,, J t such that QJ = stdevacpw T, where evac stands for the evacuation operation The tableau-pair K, Q of conjugate shapes with K a key and the frank words with t columns are characterized as follows Theorem 25 Let K, Q, with K [t], Q [n], be a pair of tableaux of Q J conjugate shapes such that evk = σm and σ S t Let, w K correspond by RSK to the pair K, Q The following statements are equivalent a K is the key associated with σ and l t,, l b J is the frank word of shape σm rev in the class of Q c stdk T [evk]m QJ = evacqj, where J is the frank word of shape σm in the class of Q J is called a frank word associated with w The frank words J with t columns and shape σm rev are those associated with some w Kσ, l t,, l with l t > 0 We give now an interpretation of the RSK correspondence in terms of compact skew-tableaux Given a word w [t] with evaluation m,, m t, let T be a skew-tableau in the compact form with word w and skew-shape

9 KEYS, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX 9 f,, f n = w, w n such that w = w w n, w i V {λ} Define the biword π π Σ = k f n = f n 5 x x k w w n Then the top word π π 2 π k = f 2 f2 n f n is such that π j is the column index, counting from left to right, of the letter x j in T, j k Thus, πj the billeter means that the letter x x j is placed in the column π j of j T For each i in [t], let J i = y i > > ym i i [n] defined by the indices of the columns of the m i letters i in T The columns words J,, J t are said the indexing sets of T and, as we have just seen, each J i records the indices of the columns where the m i letters i of w are placed, with respect to the planar representation of T Indeed we have another biword where Ji i m i Σ = Jt J 2 J t m t 2 m 2 m, 6 is the biword with bottom word i m i and top word the column J i = y i yi m i For example, the biwords Σ and Σ of the skew-tableau 2 are, respectively, Σ = and Σ = Regarding this analysis, we write often T = a, Σ = a, Σ or a, Π with Π any other equivalent biword with Σ Certainly skew-tableaux with the same compact form are characterized by the same class of biwords We are now in conditions to introduce another definition of skew-tableau which relates the combinatorial and matrix settings Given J [t], we define the characteristic function of J by χ J i =, if i J, and χ J i = 0 otherwise Given a skew-tableau T = a, Σ, we may associate the sequence of partitions a 0, a,, a t by setting a 0 := a and a i := a i + χ J i, i =,, t Clearly, each a i = a i,, ai n is a partition and satisfy a i l a i+ l a i l +, 8

10 0 O AZENHAS AND R MAMEDE for i = 0,,, t, and l =,, n Conversely, any sequence of partitions a 0, a,, a t satisfying 8 gives rise to a skew-tableau T with biword defined by the sets J i = {l : a i l = ai l +}, i =,, t For instance, the skewtableau 2 is defined by the sequence of partitions T = a 0,, a 4, where a 0 =4, 4, 2, 2, 2, a =4, 4, 3, 3, 2,,a 2 =4, 4, 4, 3, 2, 2,a 3 =5, 5, 4, 3, 2, 2 and a 4 = 6, 6, 4, 3, 3, 2 From this and theorem 24 we have Theorem 26 The RSK correspondence defined above sets up a one-to-one correspondence between pairs P, Q, with P [t], Q [n], of tableaux of conjugate shapes and a [0, 2] biwords Σ Σ b [0, 2] n t 0- matrices, where the entry i, j is iff i j is a biletter of Σ Σ c skew-tableaux in the compact form with n columns and word in [t] In the case of a tableau-pair P, Q of conjugate shapes, with Q a Yamanouchi tableau, we have Corollary 27 Let P, Q, with P [t], Q [n], be a pair of tableaux of Q J conjugate shapes Let, correspond by RSK w P to the pair P, Q The following statements are equivalent a w = P is of shape b = evq b evq = P = b c Q is the Yamanouchi tableau of evaluation b d J is a Yamanouchi word of evaluation b e J = J t J is such that J,, J t are the idexing sets of P In particular, if P = Kσ, l t,, l, the indexing sets of P are defined by the columns of the frank word [m t ] [m ] congruent with the Yamanouchi tableau of shape m If l t = 0, some m i = 0 Since there is a bijection between biwords Σ and Σ and n t 0- matrix A = a yi, where the entry a yi = exactly when the biletter y i occurs in the biword, we may represent them in a lattice of points of N 2 according to the bijection y i y, i N 2 such that y J i, i t In drawing such a lattice of points, we shall adopt the convention, as with matrices, that the first coordinate, the row index, increases as one goes downwards, and the second coordinate, the column index, increases as one goes from left to right

11 KEYS, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX The points y, i, y J i, i t, in this lattice are said the vertices of the biwords Σ, Σ or any equivalent biword For example, the vertices of the biwords 7 are represented in the following grid: The word w is read, in the grid, along rows, from right to left, starting in the top downwards to the bottom one, and the word J along columns, from bottom to top, starting in the rightmost one to the left one Compare with 2 3 The keys associated with S 4, their associated frank words and graphical representation 3 Keys associated with S 4 and their associated frank words The algorithm in section 4 as well as previous algorithms for matrix realizations of pairs of tableaux [2, 4, 6, 7] have been based on frank words σq which are the union of rows whose lengths define the conjugate shape of Q Frank words with two, three columns or, in general, for certain permutations defining the shape, this property holds In the case of four columns, there are frank words of shape 2,,, 2 which can not be splitted into a union of one row of length four with one of length two However, those frank words satisfy a row condition which enables us to stretch them to a row of length six Regarding the RSK correspondence, this phenomenon is related with words which in the shuffle of the columns of a key Let w = x x k [t] and let I be a subset of [k] We denote by w I the word x i x il, if I = {i < i 2 < < i l } Such a word w I is called a subword of w Given q words u,, u q [t] of lengths k,, k q, respectively, put k = k + + k q and let [k] = q j= I j, where I,, I q is a q-tuple of pairwise disjoint subsets of [k] with I j = k j, j =,, q The word w I,, I q defined by w I j = u j, for j =,, q, [, 8], is called a shuffle of u,, u q The words u,, u q are said the shuffle components of

12 2 O AZENHAS AND R MAMEDE w I,, I q Notice that we may have w I,, I q = w J,, J q, with J,, J q another q-tuple in the conditions above The shuffle of q words u,, u q is the set Shu,, u q = {w I,, I q : q j= I j = [k], I j = k j, w I j = u j, j [q]}, where I,, I q is a q-tuple of pairwise disjoint subsets of [k] Given a multiset A = {u,, u q } [t], we put ShA = Shu,, u q If C is another multiset, we put ShA, C = ShA C Let σ S t, t, and l t,, l a sequence of nonnegative integers For i =,, t, let R l i σ,i be the multiset defined by l i columns r σ,i The shuffle of the columns of Kσ, l t,, l, ShR l t σ,t,, R l σ,, is a subset of its plactic class [5, 7] To characterize the frank words associated with these words described by the shuffle operation, we introduce the notion of union of frank words Definition 3 Let I, J be multisets in [k] with its elements by nonincreasing order, and let x, y [n] be frank words with x = I and y = J such x y that the biword has no repeated billeters By sorting the biletters, I J consider the transformation x y Σ I J Jk J = k m k m We say that the word x I y J := J k J is the [I, J]-union of x, y For instance, the [32, 2]-union of the frank words 244 and 3 is A word congruent with Kσ, l t,, l is in the shuffle of its columns iff any associated frank word J t J is the [r σ,t l t,, r σ, l ]-union of l j row words of length j, j t, that is, J t J = t j= l j i= Ij,i r σ,j, for some rows I j,i with length j, for i =,, l j, j =, t [7] This means that, for k = t,,, there are sets A k σi J σi \ A k σi A σi A0 σi, with A 0 σi :=, such that Ak σi = l k, for i =,, k, and A k σi Ak σj, if σj < σi [, 2, 3, 4, 6]

13 KEYS, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX 3 In general, ShR l t σ,t,, R l σ, is a proper subset of the plactic class of the key Kσ, l t,, l By imposing conditions on σ S t and on the multiplicity sequence l t,, l the following result allows us to check whether the plactic class of a key either is the shuffle of its columns or not Theorem 3 [5] Let σ S t and l t,, l, l t > 0 The plactic class of Kσ, l t,, l is ShR l t σ,t,, R l σ, if and only if, for k = 2,, t with l k > 0 and r σ,k = a k a, the difference a k a is at most k Whenever a permutation σ satisfy the conditions of this theorem the same is true for rev σ Therefore, the plactic class of a key associated with the identity or with the reverse permutation in S t, t, or with any permutation in S t, t 3, coincides with the shuffle of its columns and thus associated frank words are union of rows with length given by the shape of the key In S 4 we find the first examples where the shuffle of the columns of an associated key is not all the plactic class For the permutations σ {423, 432, 423,432} we may describe the plactic class of any associated key with the shuffle operation, by adding, in those cases where the columns of the key are not enough, the single word 4342 Denote by R n 5 5 the multiset consisting of n 5 words r 5 := 4342 Kσ,, 0,, 0 Theorem 32 [5] Let l 4,, l be a sequence of nonnegative integers a Let σ S 4 The plactic class of Kσ, l 4,,l does not coincide with ShR l 4 σ,4,,r l σ, if and only if σ {423, 432, 423,432} and l 2, l 4 > 0 b Let σ {423, 432, 423,432} S 4 The plactic class of the key Kσ, l 4,, l is the union of the sets Sh R n 5 5, Rn 4 σ,4,, Rn σ,, where 0 n 5 min{l 2, l 4 }, n i = l i, i =, 3, and n i = l i n 5, i = 2, 4 To describe the frank words with four columns, that is, the frank words associated with words congruent with keys associated with S 4 and multiplicity l 4,, l, l 4 > 0, it remains to characterize the frank words associated with r 5 = 4342 Kσ,, 0,, 0 From theorem 25 we have Proposition 33 Let J = dabef c be a word The following conditions are equivalent a J is a frank word associated with 4342 b J satisfies a b c < d e f

14 4 O AZENHAS AND R MAMEDE If w Sh R n 5 5, Rn 4 σ,4,, Rn σ,, n 5 > 0, define a biword Σ as in 3, and fix a shuffle decomposition of w = w X n 5 5,, X 5,, Xn,, X, where X n 5 5,, X 5,, Xn,, X is a n + + n 5 -tuple of pairwise disjoint subsets of [ 4 i= in i + 6n 5 ], with w Xj i = r σ,j, i [n j ], j [4], and w X5 i = r 5, i [n 5 ] Denote by v the first row of Σ and let v Xj i = Ij,i, i [n j ], j [5] Then, I j,i must be a row with length j, for j [4], and we must have I 5,i = a i b i c i d i e i f i with a i b i c i < d i e i f i, for i [n 5 ] I j,i r σ,j, i [n j ], j [4], and Therefore, Σ is a shuffle of the biwords I 5,i, i [n r 5 ], and we may consider the equivalent biword 5 Π := I 5,n 5 I 5, I 4,n 4 I 4, I,n I,, 0 r 5 r 5 r σ,4 r σ,4 r σ, r σ, with bottom row the word r 5 n 5 r σ,4 n 4 r σ, n In general, there is more than one biword Π associated with Σ, each corresponding to a different shuffle decomposition of w in Sh R n 5 5,, Rn σ, On the other hand, sorting the biletters of Π by nondecreasing rearrangement for the antilexicographic order with priority of the first row, we get the biword Σ Notice that this amounts I j,i I 5,i to shuffle appropriately the biwords, j 4, and of r σ,j r 5 Π Sorting the biletters of Π by nonincreasing rearrangement for the lexicographic order with priority of the second row we get Σ This rearrangement transforms the biword into := i b i e i f i c i I 5,i Ĵi d i a r 5 r I Thus, Σ 5,i is also obtained from Π by transforming each factor into r σ,5 Ĵi r and, and then shuffling appropriately the biwords Ĵi : r I j,i r σ,j, j 4, Ĵn 5 Ĵ Π I 4,n 4 I 4, I,n I, Σ r r r σ,4 r σ,4 r σ, r σ, Proposition 34 Let σ S t, l 4,, l, l 4 > 0 a sequence of nonnegative integers, and m = 4 i= l i,, l 4 + l 3, l 4 Then

15 KEYS, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX 5 a If σ {423, 432, 423, 432}, J 4 J 3 J 2 J with shape σm rev is frank iff there exist rows I j,i of length j, j [4], i [n j ], and words Ĵ i = d i a i b i e i f i c i, with a i b i c i < d i e i f i, i [n 5 ], such that J 4 J = n 5 i=ĵi r 4 j= n j i= Ij,i r σ,j, where r = 4432, and 0 n 5 min{l 2, l 4 }, n i = l i, i =, 3, and n i = l i n 5, i = 2, 4 c J 4 J 3 J 2 J with shape σm is frank iff there is a set decomposition of the columns according to the following diagram, ABCD EFG H I ABCD J E FG ABCD H I E FG J ABCD H I E FG J ABCD H I ABCD EFG H I J ABCD EFG TUYZ H I E FG J H I X V ABCD ABCD ABCD J EFG J E FG ABCD EFG H I EF G H I J H I H I ABCD J J J TUYZ EF G ABCD ABCD H I TUYZ X EFG E FG H I ABCD H I E FG X V H I J ABCD J V ABCD EFG EF G J H I H I J J J ABCD EFG ABCD ABCD H I EFG TUYZ J H I EF G ABCD J H I EF G ABCD X V H I J J EFG ABCD H I EF G ABCD J H I EFG J H I J where A B C D, E F G, H I, and T U V < X Y Z, with A = B = C = D, E = F = G, H = I, T = U = V = X = Y = Z, and in each column the sets are pairwise disjoints We have outlined the frank words with shape σm, for each permutation σ {423, 432, 423,432}, by linking them with double lines When l 2, l 4 > 0, some of them are not only union of rows of length j, j 4, but also

16 6 O AZENHAS AND R MAMEDE of frank words of shape 2,,, 2, which can not be splitted into a union of X rows of length four and two, giving rise to the component T U Y Z V 32 Graphical representation of words in Sh R n 5 5,, Rn σ, and their associated frank words Let w Sh R n 5 5,, Rn σ,, with σ a permutation in {423, 432, 423,432}, and consider a biword Σ, as in 3 Fix a shuffle decomposition for w and consider the correspondent biword Π equivalent to Σ I 5,n 5 I Π = 5, I 4,n 4 I 4, I,n I,, r 5 r 5 r σ,4 r σ,4 r σ, r σ, where each I j,i is a row with I j,i = j, i [n j ], j [4], and I 5,i = a i b i c i d i e i f i with a i b i c i < d i e i f i, i [n 5 ] Consider the graphical representation of the vertices of Π Linking, by a I j,i r σ,j and straight line, the vertices of consecutive biletters of each factor I 5,i, we get a graphical representation of each shuffle component of w r 5 Therefore is graphically represented by n n polygonal lines, n j polygonal lines of nonnegative slope corresponding to the shuffle component I j,i I 5,i, i [n r j ], j [4], and n 5 to, i [n σ,j r 5 ], where the line linking 5 the vertices c i and d i 4 have negative slope Example 3 Consider the biword Σ 7, whose bottom word is a shuffle of 4342, 432 and 4 We may sort the billeters of Σ in several ways, in order to obtain a biword Π Take, for instance, the biword Π =

17 KEYS, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX Linking, respectively, the vertices of the consecutive billeters of and by a polygonal line, we get the following graphical representation of the bottom word of Σ where the underlined letters 432 indicate the shuffle component 432, and the upperlined letter indicates the shuffle component 4: Since the shuffle components r σ,j, j =,, 4, are columns, the rightmost letter of r σ,j corresponds, in the graphical representation, to the leftmost vertex of its polygonal line By the leftmost vertex of a polygonal line of r 5 = 4342, we mean the vertex in column, in that polygonal line, having the biggest row value, which corresponds to the rightmost letter of r 5 For instance, in 3, the leftmost vertex 6, of the polygonal line of 4342 corresponds to the rightmost letter of this shuffle component Two vertices a, b, x, y of Π are linked if they are consecutive vertices of a polygonal line In this case, if b < y and a x, a, b is said positively-linked to x, y, and x, y is said negatively-linked to a, b For instance, in the graphical representation 3, the vertex 5, 4 is negatively-linked to 6, 2, but is not positively-linked to any vertex Definition 32 Let u, v be two shuffle components of w A vertex b, y u is said a critical vertex of {u, v} if one of the following conditions holds: i b, y is negatively-linked to a vertex b, y with y y >, or it represents the right most letter of u, and there is a pair of linked vertices b, x, a, y in v, with y > x y ii b, y is positively-linked to a, x and there is a vertex a, y in v Notice that if b, y u is a critical vertex satisfying condition i above, then we must have b = 4 or b = 3, and u is either r σ, = 4, or r σ,2 = 4, or r σ,3 = 42, or r σ,3 = 43, or r 5 = 4342

18 8 O AZENHAS AND R MAMEDE Example 32 In 3, the vertex 4, 432 is a critical vertex of {4342, 432}, since it is positively-linked to 3, 2 and there is a vertex 3, Another example is given by the biword, where 2, is a critical vertex, since it is negatively-linked to 3,, with 3 >, and the vertices 2, 2 and, 3 are linked: A word w in Sh R n 5 5,, Rn σ,, has, in general, several shuffle decompositions Given a shuffle decomposition of w, it is a simple task to adjust, if necessary, the links between the vertices of the biword Π, and, therefore, the shuffle decomposition itself, in order to form a new biword Π satisfying the conditions of the following lemma Lemma 35 There is a shuffle decomposition of w such that the corresponding biword Π satisfies the following conditions: a Any two shuffle components have no critical vertices I 5,i abcdef b If = is a shuffle component of Π, then: r i in row c there is, at most, one more vertex, placed in column 4, which must be negatively-linked to a vertex in column 3; ii if d e, in row d there is, at most, one more vertex, placed in column, which must be positively-linked; iii if d e, in row e, to the right of e, 2, there are no vertices Proof: We prove only condition biii All other conditions are proven in a similar way Recall that we must have a b c < d e f If e, 3 is also a vertex of Π, then it must belong to one of the words 4342, 432 or 43 In these cases, Π must have a sub-biword either of the form α = abcdef gehjkl, or β = abcdef gehi or γ = abcdef geh,

19 KEYS, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX 9 with g e h i < j k l We may re-link the vertices of these shuffle components by replacing in Π the biword α, β or γ with the biword abcjkl geef dh abcdhi geef geef abc dh, or, respectively In any case, the vertex e, 2 belongs now to the new shuffle component 432, and is linked to e, 3 Notice also that we have only changed the links between the two shuffle components Therefore, we may assume, without loss of generality, that if e, 2 is a vertex of a shuffle component r 5, there are no vertex in position e, 3 Assume now that e, 4 is a vertex, but e, 3 is not An analysis similar to the one done above shows that if e, 4 represents one of the letters 4 of 4342, or belongs to the columns 432, 43, 42, 4 or 4, then we may re-link the vertices of these shuffle components in such a way that the vertex e, 2 is linked to e, 4 Therefore, we may assume that e, 4 is not a vertex of Π Example 33 The biword Π 2, graphically represented in 3, fails to satisfy condition bi of the lemma above, since the middle letter of 4342 is represented by the vertex 3,, and there is a vertex in row 3, column 2 Rearranging the links between the vertices, and therefore the shuffle decomposition itself, we obtain the biword Π = all the required conditions of lemma , which satisfy Example 34 In example 32, the biword Π = does not satisfy condition i, since 2, 3 is a critical vertex Rearranging the links between the vertices, we obtain the biword Π =, represented

20 20 O AZENHAS AND R MAMEDE below, which satisfy all the required conditions of lemma 35: In what follows, we assume that our shuffle decomposition of w satisfies the conditions of lemma 35 Definition 33 [7] Consider the biword Π For each vertex a, b of Π, define the map [0, b] [, a] n s n a,b defined as follows: I If there is no vertex of Π in row a to the left of column b, then let s n a,b := a II Otherwise, let a, b, b < b, be the rightmost vertex of Π, in row a, to the left of a, b a If b n, put s n a,b := a b If b > n and a, b is positively-linked to a vertex x, y of Π, with x < a, put s n a,b := sn x,y c Else, put s n a,b := sn a,b The number s n a,b, with n < b, indicates, according to a certain path, a row x a with a vertex x, y, n y b, such that there are no vertices in the interval ]n, y[ Since by lemma 35 a, there are no critical vertices in our fixed shuffle decomposition, we find that s n a,b a only if b = 4 and a, 4 is either negatively-linked to a vertex in column or it corresponds to the shuffle component r σ, = 4 For instance, consider the biword Π displayed in example 33 Since the vertex 6, is placed in column, by rule I we have s 0 6, = 6 By the same reason, s 0 3, = 3 To compute s0 3,4, note that 3, 2 is the vertex closest to 3, 4, and it is positively-linked to 2, 3 Thus, by rule IIb, s 0 3,4 = s0 2,3 = 2, since there are no vertices to the left of this last vertex We have s 3,4 =, by rule IIa, and s0,4 = s0,3 =, by IIc and I

21 KEYS, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX 2 Corollary 36 a If a, i is the leftmost vertex of a shuffle component r σ,k, then s 0 a,i a only if σ {423, 432}, i = 4, k =, and the nearest vertex in row a is in column, representing the underlined letter of 4342, 432, 43 or 42, or in column 2, representing the underlined letter 2 of 432 b If b, j, a, i are linked vertices of a shuffle component r σ,k, then s j a,i a only if j =, i = 4, and the nearest vertex in row a is in column 2, representing the underlined letter 2 of 432 Proof: Follows from lemma 35 a 33 An injection of the plactic class of a key associated with S 4 into S 6 For each σ = σσ2σ3σ4 S 4, define the permutation σσ2σ3σ434 S 6, where { σk, σk =, 2 σk = σk + 2, σk = 3, 4 This correspondence is a bijection between S 4 and the set {σ = α34 S 6 : α a permutation on {256}} S 6 In particular, the set S := {423, 432, 423, 432} is transformed into {62534, 65234, 62534, 65234} Let ρ : { r 5, r σ,j : j [4]} {r σ,j : j [6]}, such that ρ r 5 = r σ,6 and ρr σ,j = r σ,j, j =, 2, 3, 4 Given σ S 4 and a sequence l 4,, l of nonnegative integers with l 4 > 0, [Kσ, l 4,, l ] := {w [4] : w Kσ, l 4,, l } = { ShR n 4 σ,4 =,, Rn σ,, if σ S 4 \ S, or σ S and l 2 = 0; min{l2,l 4 } n 5 =0 Sh R n 5 5, Rn 4 σ,4,, Rn σ,, if σ S and l 2 0, where n i = l i, i =, 3, and n i = l i n 5, i = 2, 4 For each n 5 = 0,, min{l 2, l 4 }, the map ρ can be extended, by shuffling, to a bijection between Sh R n 5 5, Rn 4 σ,4,, Rn σ, and Sh R n 5 σ,6, Rn 4 σ,4,, Rn σ, [Kσ, n 5, 0, n 4, n 3, n 2, n ] Thus every word in the plactic class of the key Kσ, l 4,, l has a copy in the shuffle of the columns of some key, associated with S 6 Let w Sh R n 5 5, Rn 4 σ,4,, Rn σ,, and Σ a biword with bottom row w Fix a shuffle decomposition satisfying the conditions of lemma 35 and consider

22 22 O AZENHAS AND R MAMEDE the correspondent biword Π 0 Let Π = I 5,n 5 I 5, I 4,n 4 I 4, I,n I, r σ,6 r σ,6 r σ,4 r σ,4 r σ, r σ, 5 I j,i be the biword obtained by transforming each shuffle component and r σ,j a i b i c i d i e i f i I j,i a of Π, into and i b i c i d i e i f i respectively, where a r σ,j i b i c i < d i e i f i Let Σ be the biword obtained by sorting the biletters of Π by nondecreasing rearrangement for the antilexicographic order with priority on the first row The second row of Σ, denoted ρw, is a shuffle of columns r σ,j, j = 6, 4, 3, 2, After this injection the frank word d i a i bi ei fi c i, satisfying a i b i c i < d i e i f i, associated with 4342, is stretched to a row word a i b i c i d i e i f i associated with ρ4342 = Thus the frank word J 4 J 3 J 2 J with four columns having d i a i b i e i f i c i as a component is transformed into one of six columns J 6 J 5 J, now a union of rows having the row a i b i c i d i e i f i as a component, such that J = J \ {c i : i [n 5 ]}, J 2 = J 2, J 5 = J 3, J 3 = {d i : i [n 5 ]}, J 4 = {c i : i [n 5 ]}, and J 6 = J 4 \ {d i : i [n 5 ]} Example 35 Let σ = 423, consider the biword Σ 7, whose bottom word is a shuffle in Sh r 5, r σ,4, r σ,, and the biword Π = graphically represented in example 33 The frank word associated with w is a [442, 432, 4]-union of the frank words , 2233 and 3 We have σ = Applying the map ρ, we obtain the biword

23 KEYS, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX 23 Π = , whose graphical representation is given below The bottom word of the biword Σ associated with Π is ρw = , a shuffle of the columns 65432, 652 and 6, precisely the words read along the polygonal lines, now all having nonnegative slope The frank word associated with ρw is now a [65432, 652,6]-union of row frank words 4566, 2233 and 3 In the next proposition, we state some properties of the biword Π abcdef Proposition 37 Let Π be the biword defined above, and one of its shuffle components Then, a Π has no critical vertices b There are no vertices neither in row c to the left of c, 4, nor in row d to the right of d, 3, nor in row e to the right of e, 2 Proof: It is a consequence of the definition of Π and lemma 35 We are now ready to get into the matrix framework 4 Matrix realizations of pairs of tableaux 4 An algorithm and statement of results Let R p be a local principal ideal domain with maximal ideal p, and let U n be the group of n n unimodular matrices over R p All the matrices in this paper are n n nonsingular matrices with entries over R p Given a matrix A, A T will denote the transpose of A Given matrices A and B, we say that B is left equivalent to A written B L A if B = UA for some unimodular matrix U; B is right

24 24 O AZENHAS AND R MAMEDE equivalent to A written B R A if B = AV for some unimodular matrix V ; and B is equivalent to A written B A if B = UAV for some unimodular matrices U, V The relations L, R and are equivalence relations in the set of all n n matrices over R p Let A be an n n nonsingular matrix By the Smith normal form theorem [8, 7], there exist nonnegative integers a,, a n with a a n such that A is equivalent to diagp a,, p a n The sequence a = a,, a n, of the exponents of the p-powers in the Smith normal form of A, is a partition of length n, uniquely determined by the matrix A We call a the invariant partition of A More generally, if we are given a sequence of nonnegative integers e,, e n, the following notation for p-powered diagonal matrices will be used: diag p e,, e n := diagp e,, p e n Given a subset J [n], we put D J := diag p χ J We denote by E ij the n n matrix having in position i, j and 0 s elsewhere, and define the elementary unimodular matrices T ij x as follows: T ij x = I + xe ij, where i j and x R p ; T ii v = I + v E ii, where v is a unit of R p It is obvious, that E ij E rs = δ jr E is, where δ jr denotes the Kronecker symbol, that is, δ jr = if j = r, and equals 0 otherwise In the lemma below we state some basic properties of the these elementary matrices T ij x, which will be used later Lemma 4 Let i, j, r, s, m [n], and x, y, v R p, such that v is a unit Then, i T ij xt rs y = T rs yt ij x, whenever i s and j r ii T ij xt js y = T js yt ij xt is xy, if i s iii T ii vt rs x = T rs uxt ii v, for some unit u iv T ij xd [m] = D [m] T ij x, if i, j / [m] v T ij xd [m] = D [m] T ij xp, if i / [m] and j [m] Proof: Straightforward Given a partition a, let a := diag p a

25 KEYS, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX 25 Theorem 42 Let U U n, and m n Given a partition a of length n, there exist J [n], with J = m, and V U n such that a UD [m] R a V D J L a D J = diag p a + χ J, with a + χ J a partition Proof: See the proof of theorem 37 in [6] We recall now the discussion in section 22 and put it into our matrix framework Given a sequence of n n nonsingular matrices B,, B t, where B r has elementary invariant partition m k, for k =,, t, there exist U,, U t U n such that B B k R U D [m ]U 2 D [m2 ]U k D [mk ], for k =,, t Using previous theorem, we also find matrices V 2,, V t U n, and sets F 2,, F t [n], with F i = m i, 2 i t, such that, for k =,, t, B B 2 B k R U D [m ]U 2 D [m2 ] U k D [mk ] R U D [m ]V 2 D F2 V k D Fk L D [m ]D F2 D Fk = diag p χ [m ] + χ F χ F k, 7 with b k := χ [m] + χ F χf k a partition Therefore, we may assume without loss of generality, that our sequence B,, B t has the form V D F,, V t D Ft, 8 where V,, V t U n and F := [m ], F 2,, F t [n], with F i = m i, i i t, such that diag p χ [m] + χ F χ F k is a partition, k =,, t The sets F,, F t are uniquely determined by the sequence B, B 2,, B t In particular, when F i = [m i ], i t, it has been proved [2, 4, 6] that we may consider in 8, V 2 = = V t = I n In general, however, this is not the case For instance, consider the sequence B, B 2, B 3, where B := D {}, B i := D {2}, i = 2, 3, and note that B B 2 B 3 D {} D {2} D {} Assume the existence of an unimodular matrix U U 2 such that i UD {} R D {}, ii UD {} D {2} R D {} D {2} and iii UD {} D {2} D {2} R D {} D {2} D {} By theorem 35 in [6], we may write U = T n P σ QL, for some σ S 2, where L is lower triangular with units along the main diagonal, Q is upper triangular with [ s ] along the main diagonal and multiples of p above it, and T n = xp n, for some n 0 Since UD 0 {} R D {}, we must have σ = id and n > 0 But then, UD {} D {2} D {2} R T n D {} D {2} D {2} R D {} D {2} D {2} R D {} D {2} D {},

26 26 O AZENHAS AND R MAMEDE contradicting condition iii Hence there is not a matrix U U 2 satisfying conditions i, ii and iii, The sequence 7 gives rise to the biwords Σ B = Ft F t m t m b n Σ B = b n, with w w w w n a tableau of shape b t = b,, b n n and indexing sets F,, F t, and corresponds by RSK to the tableau-pair P B, Q B of conjugate shapes, where P B = w w n [t] and Q B = PF t F [n] the Yamanouchi tableau of evaluation P B Let a be a partition, and consider now the sequence a, B,, B t Using 7, 8, we may assume without loss of generality, that the sequence a, B,, B t has the form a, V D F, V 2 D F2,, V t D Ft 9 Using again theorem 42, there exist matrices U,, U t U n and sets J,, J t [n], with J i = F i = m i, i t, such that a B B k R a V D F V 2 D F2 V k D Fk R a U D J U 2 D J 2 U k D J k L diag p a + χ J + χ J χ J k, with a + χ J + χ J χ J k a partition, for k =,, t The sequence 9 gives rise to the skew-tableau T := a, Σ Jt J = t m t m and corresponds by RSK to the tableau-pair P, Q of conjugate shapes, where Q = PJ t J and P is the tableau congruent with the word of T, and simultaneously to the pair P B, Q B of tableaux of conjugate shapes, where P and P B have the same evaluation m,, m t and Q B is the Yamanouchi tableau of evaluation P B We have proved in [5] that when 7 has the form UD [m ] D [mt ], then P = P B is the key of evaluation m,, m t, and Q B is the Yamanouchi tableau of shape m,, m t by nonincreasing order Thus the sequence a UD [m ] D [mt ] gives rise to the pairs P, Q and P, Q B of tableaux of conjugate shapes such that P is the key of evaluation m,, m t and Q B the Yamanouchi tableau of shape m,, m t by nonincreasing order The key of evaluation m,, m t is a leaf of the rooted tree of all tableaux of evaluation m,, m t In general, we do not have P = P B, as we may

27 KEYS, FRANK WORDS AND MATRIX REALIZATIONS OF PAIRS OF TABLEAUX 27 observe in the next example, where P B is the root of the tree, the unique row tableau of evaluation m,, m t Example 4 a Let B = P 24 P 35 D {,2,3}, B 2 = D {4,5} The sequence B B 2 L D {,2,3} D {4,5} gives rise to the biword Σ B =, and 22 hence to the pair P B, Q B, where P B = 22 On the other hand, the sequence a B B 2 R a D {,4,5} D {2,3} leads to the skew-tableau a, Σ = 32 54, and corresponds to the tableau-pair P, Q, where P = b Let B = P 34 D {,2,3}, B 2 = D {4,5} The sequence B B 2 L D {,2,3} D {4,5} gives rise to the biword Σ B =, and to the pair P 22 B, Q B, where P B = 22 On the other hand, the sequence a B B 2 R a D {,2,4} D {3,5} leads to the skew-tableau a, Σ = and corresponds to the 22 tableau-pair P, Q, with P = 2 2 c Let B = D {,2,3}, B 2 = D {4,5} The sequence B B 2 = D {,2,3} D {4,5} gives rise to the biword Σ B =, and to the pair P 22 B, Q B, where P B = 22 But the sequence a B B 2 = a D {,2,3} D {4,5} gives the skewtableau a, Σ = and corresponds to the tableau-pair P, Q, with P = 22 = P B Therefore the sequences a, UD {,2,3}, D {4,5}, with U running over U 5, give rise to skew tableaux with words congruent with P running over the set {22; 2 2; 2 2 } the set of all tableaux of evaluation 3, 2 This example can be easily generalized to all tableaux of evaluation m, n, with m n The sequences a, UD {,,m}, D {m+,,m+n}, with U running over U m+n, give rise to words congruent with P running over the tableaux of evaluation m, n Regarding the RSK correspondence between skew-tableaux and pairs of tableaux of conjugate shapes, following [2, 4, 6], we introduce the definition of a matrix realization of a pair of tableaux T, F with T a skew-tableau and F a tableau with the same evaluation as T

28 28 O AZENHAS AND R MAMEDE Definition 4 Let P, Q and P B, Q B, P, P B [t], Q, Q B [n], be two pairs of tableaux of conjugate shapes such that evp = evp B = m,, m t, and Q B is the Yamanouchi tableau of evaluation P B Let T = a 0, a,, a t be a skew-tableau which corresponds by RSK to P, Q and let P B = 0, b,, b t We say that a sequence of n n nonsingular matrices A 0, B,, B t is a matrix realization of [P, Q; P B, Q B ] or T, P B if: I For each r {,, t}, the matrix B r has invariant partition m r II For each r {,, t}, the matrix B B r has invariant partition b r III For each r {0,,, t}, the matrix A r := A 0 B B r has invariant partition a r The pair [P, Q; P B, Q B ], or T, P B, is called admissible For the purpose of this paper, we shall consider only the pairs P, Q and K, Y of tableaux of conjugate shapes with evp = evk such that K is the key of evaluation m,, m t and Y is the Yamanouchi tableau of shape m,, m t by nonincreasing order Thus, in order to verify property II, it is sufficient to show that B B t has invariant partition m + + t m t = K Given the sequence m,, m t of nonnegative integers, the tableau with this evaluation and shape t i= m i is the key K = 0, m, m + m 2,, t i= m i Therefore, definition 4 becomes: Definition 42 Let P, Q and K, Y, K [t], Q [n], be two pairs of tableaux of conjugate shapes such that evp = evk, K is the key of evaluation m,, m t and Y the Yamanouchi tableau of shape m,, m t by nonincreasing order Let T = a 0, a,, a t be a skew-tableau which corresponds by RSK to P, Q We say that a sequence of n n nonsingular matrices A 0, B,, B t is a matrix realization of [P, Q; K, Y] or T, K if: I For each r {,, t}, the matrix B r has invariant partition m r II The matrix B B t has invariant partition K III For each r {0,,, t}, the matrix A r := A 0 B B r has invariant partition a r It has been shown in [5] that [P, Q; K, Y] is an admissible pair only if P = K Let σ S t and l t,, l be a sequence of nonnegative integers Let L t+ :=0 and L k := L k+ + l k, k =,, t The next algorithm, described in [7], gives a procedure to obtain a matrix realization for the pair T,