The recording tableaux of the quantum Littlewood-Richardson map and the orthogonal transpose symmetry map

THE RECORDING T ABLEA UX OF THE QUANTUM LITTLEW OOD-RICHARDSON MAP AND THE OR THOGONAL TRANSPOSE SYMMETR Y MAP OLGA AZENHAS Abstract. Recen tly W atanab e has given an algorithm to compute a bijection, that he calls (quantum) Littlewood-Richardson (LR) map, betw een semi-standard Y oung tableaux of shape a partition with at most 2 n parts and pairs of tableaux consisting of a symplectic tableau with shape a partition with at most n parts, and a recording tableau of sk ew-shap e given by the tw o previous shapes. The recording tableaux in that algorithm are sho wn to b e equinumerous to Littlewoo d-Ric hardson-Sundaram tableaux whose injectivit y is shown com binatorially while the surjectivit y is concluded via represen tation theory of a quantum symmetric pair of t ype AI I. Henceforth, the algorithm to compute the quan tum LR map pro vides a new branching model for the branc hing multiplicities from GL 2 n ( C ) to Sp 2 n ( C ). Here, as morally suggested b y W atanab e, one provides a com binatorial pro of of the surjectivity of the quan tum LR map which in turn exhibits the restriction of the LR orthogonal transpose symmetry map to LR-Sundaram tableaux. Contents 1. In tro duction 1 Ac knowledgemen ts 4 2. Preliminaries 4 2.1. Sc hensted column insertion and rev erse 5 3. The linear time bijection betw een the sets e Rec 2 n ( λ/µ ) and LR-Sundaram tableaux LRS 2 n ( λ/µ ) 6 3.1. The restriction of the LR orthogonal transpose linear time map to LR-Sundaram tableaux 9 4. The con tainmen t of the set e Rec 2 n ( λ/µ ) in to the set of recording tableaux Rec 2 n ( λ/µ ) 10 4.1. Symplectic tableaux 10 4.2. Reduction map 11 4.3. Expanding map b y rev erse Sc hensted column insertion and rev erse remo v als 13 References 20 1. Introduction The Littlewoo d-Ric hardson map b y W atanab e [W at25b] can be seen as a generalization of the branc hing rule from GL m ( µ ) × GL m ( ν ) to GL m ( λ ). More precisely , a generalization of the G. Thomas [Tho78] bijection on S S T m ( µ ) × S S T m ( ν ) to S S T 2 n ( λ ). F or the sake of comparison, G. Thomas bijection asserts S S T m ( µ ) × S S T m ( ν ) ∼ − → G λ ∈ P ar ≤ m S S T m ( λ ) × LR ( λ/µ, ν ) . where the recording tableau of the Sc hensted column insertion of the tableau pair ( U, V ) is stored in the form of an LR tableau in LR ( λ/µ, ν ) for some partition λ . V ery imp ortan tly is that this bijection lifts to a g l m - crystal isomorphism [Kw o09] giving sim ultaneously the crystal v ersion of the original Littlew o od-Richardson 2000 Mathematics Subject Classific ation. 05E05, 05E10, 05E14, 17B37, 68Q17. Key words and phr ases. Symplectic tableau, branc hing rule, Littlewoo d-Ric hardson map. 1 2 OLGA AZENHAS (LR) rule [Lit44] and the Berenstein-Gelfand-Zelevinsky LR rule [GZ85, GZ86, BZ88] on Gelfand-Tsetlin patterns [GT50], B ( µ, m ) ⊗ B ( ν, m ) ≃ M T ∈ LR ( λ/µ,ν ) λ ∈ P ar ≤ m B ( λ, m ) × { T } ≃ M λ ∈ P ar ≤ m B ( λ, m ) c λ µ,ν (1) whose highest and low est weigh ts of B ( λ, m ) × { T } exhibit the righ t resp ectiv ely left companions of each LR tableau T . The multiplicit y c λ µ,ν of the crystal B ( λ ) in this decomp osition shows that LR ( λ/µ, ν ) is equin umerous to the right and left companions in the form of Gelfand-Tsetlin patterns in the Berenstein- Gelfand-Zelevinsky LR rule. F rom [Kwo18, LL20, KT25, Aze26] it turns that the decomp osition (1) has a refinement to include LR- Sundaram (LRS)-tableaux. F or a fixed n ∈ N , m = 2 n , µ ∈ P ar ≤ n a partition with at most n parts, and ν, λ ∈ P ar ≤ 2 n suc h that the ν ′ the transp ose or conjugate of ν has ev en columns, LR-Sundaram tableaux LRS ( λ/µ, ν ) ⊂ LR ( λ/µ, ν ) B ( µ, n ) ⊗ B ( ν, 2 n ) ≃ M T ∈ LRS ( λ/µ,ν ) λ ∈ P ar ≤ 2 n B ( G S p µ ⊗ Y ( r evν )) × { T } G M T ∈ LR ( λ/µ,ν ) \ LRS ( λ/µ,ν ) λ ∈ P ar ≤ 2 n B ( G µ ⊗ Y ( r evν )) × { T } ≃ M G S p µ ⊗ Y ( rev ν ) ≃ Y ( rev λ ) λ ∈ P ar ≤ 2 n B ( λ, 2 n ) spc λ µ,ν G M G µ ⊗ Y ( rev ν ) ≃ Y ( rev λ ) G µ / ∈ S pT ( µ ) λ ∈ P ar ≤ 2 n B ( λ, 2 n ) c λ µ,ν where spc λ µ,ν is the cardinality of LRS ( λ/µ, ν ) or the num b er of symplectic low est w eight tableaux of shap e λ of the form G sp µ ⊗ Y ( rev ν ) ≃ Y ( r ev λ ) where G S p µ ∈ S pT ( µ ) with weigh t r ev ( λ − ν ) is a symplectic semi-standard tableau of shap e µ . Let λ ∈ P ar ≤ 2 n b e a partition with at most 2 n parts. The Littlew o od-Richardson map LR I I is an one-to-one assignment of a semi-standard tableau T of shap e λ ∈ P ar ≤ 2 n to a pair ( P I I ( T ) , P I I ( T )) for some shap e µ ∈ P ar ≤ n , consisting of a symplectic tableau in S pT 2 n ( µ ) resp ectiv ely a recording tableau in Rec 2 n ( λ/µ ), LR AI I : S S T 2 n ( λ ) ∼ − → G µ ∈ P ar ≤ n µ ⊂ λ S pT 2 n ( µ ) × Rec 2 n ( λ/µ ) . (2) F urthermore, for each µ ∈ P ar ≤ n and µ ⊂ λ , the recording tableaux Rec 2 n ( λ/µ ) in the Littlewoo d- Ric hardson map LR AI I are in natural bijection with LR-Sundaram tableaux [Sun86, Sun90] Rec 2 n ( λ/µ ) ∼ − → LRS 2 n ( λ/µ ) (3) where LR S 2 n ( λ/µ ) is the set of LR-Sundaram tableaux of shap e λ/µ (or LR symplectic tableaux LRT S p 2 n ( λ/µ ) in [W at25b]). Henceforth, the algorithm defining the map LR AI I giv es rise to a bijection from the set of semi- standard tableaux of a fixed shap e say λ to a disjoint union of several copies of sets of symplectic tableaux of v arious shap es where the multiplicit y of each shap e µ ⊂ λ equals to c λ µ the num b er of LR-Sundaram tableaux of shap e λ/µ S S T 2 n ( λ ) ∼ − → G µ ∈ P ar ≤ n µ ⊂ λ Q ∈ LRS 2 n ( λ/µ ) S pT 2 n ( µ ) × { Q } . (4) Thereb y , the recording tableaux in the algorithm defining the map LR AI I giv e a new branching rule from Gl 2 n ( C ) to S p T 2 n ( C ). It is then demanding to ha ve a complete com binatorial pro of of this new branching rule. As suggested in [W at25b] the bijections (2) and (3) might b e prov ed in the realm of com binatorics. THE RECORDING T ABLEAUX IN THE QUANTUM LITTLEWOOD-RICHARDSON MAP 3 The injectivity of the Littlewoo d-Ric hardson map (2) and the injectivity of the bijection (3) are prov ed com binatorially in [W at25b] by showing, Rec 2 n ( λ/µ ) ⊆ e Rec 2 n ( λ/µ ) ∼  → LRS ( λ/µ ) . The surjectivity of (2) and (3) is concluded from the representation theory of a quantum symmetric pair of type AI I 2 n − 1 in [W at25b, Section 7.2] resp ectiv ely [W at25b, Theorem 8.2]. The Littlewoo d-Richardson map (2) has a q analogue isomorphism by showing how the irreducible U -mo dule V ( λ ) decomp oses into irreducible U ı -submo dules at q = ∞ which allows to conclude the bijectivity of (2) LR AI I V ( λ ) → M µ ∈ P ar ≤ n V ı ( µ ) ⊗ Q Rec ( λ/µ ) Our main results assert as follows. Theorem A (Theorem 1)[The surjectivity of the map in [W at25b, Lemma 8.3.2]] Let T ∈ LRS 2 n ( λ/µ ) of ev en weigh t ν and ν t its conjugate partition. Let J 1 , . . . , J ν 1 the decomp osition of T into vertical strips where each strip is filled with 1 , 2 , . . . , | J i | = ν t i ∈ 2 Z for i = 1 , . . . , ν 1 . Let ♢ b e the map that relab els each v ertical strip J i with ν t i i ’s, for i = 1 , . . . , ν 1 . Then w e get a new tableau Q ∈ e Rec 2 n ( λ/µ ) of weigh t ν t and ♢ is a bijection b et ween LRS 2 n ( λ/µ ) and e Rec 2 n ( λ/µ ). Equiv alently , ♢ is the in verse map of the map in [W at25b, Lemma 8.3.2]. Corollary (Section 3.1) The LR orthogonal transp ose symmetry map ♦ [ACM25] restricts to LR-Sundaram tableaux ♦ : LRS 2 n ( µ, ν, λ ) ♢ − → e Rec 2 n ( µ, ν t , λ )  → π ◦ t LR ( λ t , ν t , µ t ) : T 7→ ♢ T = Q 7→ Q π ◦ t = ♦ T where Q π ◦ t means the π -rotation (transp osition) follow ed with transp osition (rotation) of the Y oung diagram D ( λ ). In other words, the bijection ♢ , and therefore the injection in [W at25b, Lemma 8.3.2], exhibits the restriction of the LR orthogonal transp ose symmetry map ♦ , c µ,ν,λ = c λ t ,ν t ,µ t , to LR Sundaram tableaux. The next theorem giv es the constructive fundamen tal step for the un winding of the sequence of successors [W at25b, Lemma 8.3.1] in the algorithm to computing the Littlewoo d-Ric hardson map. The unwind pro ce- dure goes from the last to the first successor, and the theorem giv es explicitly LR AI I − 1 ( S, Q ) for the un wind of the last successor where Q ∈ e Rec 2 n (( λ 0 ,  l − ℓ ( µ ) ) /µ ) and S ∈ S p 2 n ( µ ). The symplectic prop ert y of S in the first winding step is imp ortan t to ensure the following unwinding steps as they comprise S ∈ S S T 2 n . Eac h un wind step consists of reverse Schensted column insertion follow ed with reverse remov al. Theorem B (Theorem 2)[The inclusion Rec 2 n ( λ/µ ) ⊇ e Rec 2 n ( λ/µ ) for [W at25b, Lemma 8.3.1]] Let µ ⊆ λ 0 ∈ P ar ≤ n suc h that  ( λ 0 ) =  ( µ ) and λ 0 /µ is a vertical strip of length l 0 = | λ 0 | − | µ | . Let S ∈ S p 2 n ( µ ) and Q ∈ e Rec 2 n (( λ 0 ,  l − ℓ ( µ ) ) /µ ) and thereby l ≤ 2 n −  ( µ ) + l 0 ⇔ l +  ( µ ) − l 0 ≤ 2 n, and 0 ≤ l −  ( µ ) + l 0 ∈ 2 Z . Let r 1 < · · · < r ℓ ( µ ) − l 0 b e the complement of the set of the row co ordinates of the cells of the vertical strip λ 0 /µ in [1 ,  ( µ )]. Let S ′ 1 = ( a 1 , . . . , a ℓ ( µ ) − l 0 ) ∈ S p 2 n (  ℓ ( µ ) − l 0 ) b e the set of the bump ed elements from the first column S 1 of S by applying successively the reverse column insertion to the ro ws r ℓ ( µ ) − l 0 < · · · < r 1 (from the largest to the smallest ro w) of S , and S ′ ≥ 2 the returned tableau b y the application of that reverse column insertion on S . Then this pro cedure applied to the pair ( S, Q ) returns T = ( T 0 T 1 · · · T ℓ ( µ ) − l 0 ) .S ′ ≥ 2 ∈ S S T 2 n (( λ 0 ,  l − ℓ ( µ ) )) where T 0 = (1 , 2 , . . . l 1 ) , (5) T i = ( ( a i , a i + 1 , . . . , a i + l i +1 ) , if a i ∈ 2 Z ( a i , a i + 1 + 1 , a 1 + 1 + 2 , . . . , a i + 1 + l i +1 ) , if a i / ∈ 2 Z 1 ≤ i ≤  ( µ ) − l 0 . (6) 4 OLGA AZENHAS with l 1 = ( a 1 − 2 , if a 1 ∈ 2 Z a 1 − 1 , if a 1 / ∈ 2 Z , (7) l i +1 =                a i +1 − a i − 2 , if a i ∈ 2 Z , a i +1 ∈ 2 Z or a i / ∈ 2 Z , a i +1 / ∈ 2 Z ∧ a i +1 − a i ≥ 2 i + 1 a i +1 − a i − 1 , if a i ∈ 2 Z , a i +1 / ∈ 2 Z or a i / ∈ 2 Z , a i +1 ∈ 2 Z ∧ a i +1 − a i ≥ 2 i + 1 0 , otherwise , 1 ≤ i <  ( µ ) − l 0 . (8) and l ℓ ( µ ) − l 0 +1 = l −  ( µ ) + l 0 − ℓ ( µ ) − l 0 X i =1 l i . This pap er is organized in four sections. Section 2 introduces the relev an t notation on semistandard tableaux and recalls the Schensted column insertion and its rev erse. Section 3 prov es the surjectivity of the map e Rec 2 n ( λ/µ ) ∼  → LRS ( λ/µ ) in Theorem 1 (Theorem A in the Introduction) and how it exhibits the restriction of the LR symmetry c µ,ν,λ = c λ t ,ν t ,µ t , to LR Sundaram tableaux in Subsection 3.1, Corollary 1 in the Introduction. Section 4 recalls the relev ant op erations of the algorithm that computes the quantum Littlew o od-Richardson map and their prop erties, With this on hand we get prepared for the fundamental un wind step of the sequence of successors of that algorithm to prov e the surjectivity of the map S S T 2 n ( λ ) ∼  → G µ ∈ P ar ≤ n µ ⊂ λ Q ∈ LRS 2 n ( λ/µ ) S pT 2 n ( µ ) × { Q } , pro viding we know that e Rec 2 n ( λ/µ ) ∼ → LR S ( λ/µ ) which is done in Theorem A. This fundamental step is explicitly exhibited in Theorem 2 (Theorem B in the Introduction). A cknowledgements The author ackno wledges financial supp ort by the Centre for Mathematics of the Universit y of Coim- bra (CMUC, https://doi.org/10.54499/UID/00324/2025) under the Portuguese F oundation for Science and T echnology (FCT), Grants UID/00324/2025 and UID/PRR/00324/2025. 2. Preliminaries A p artition is a weakly decreasing sequence of nonnegative integers γ = γ 1 ≥ γ 2 ≥ · · · such that γ k = 0 for some k ≥ 1. The maximal i such that γ i > 0 is called the numb er of p arts or length of γ , denoted  ( γ ). F or each m ≥ 0, the set of partitions of length at most m is denoted b y P ar ≤ m . W e assume the inclusion P ar ≤ m ⊂ P ar ≤ k whenev er k ≥ m . Th us we often write the partition γ as a v ector γ = ( γ 1 , γ 2 , . . . , γ k ) for k ≥  ( γ ). Given l ∈ Z ≥ 0 ,  l denotes the partition of length l whose parts are all 1, that is,  l = (1 , . . . , 1 | {z } l ) =: (1 l ). The partition γ is said to b e even if γ 2 i − 1 = γ 2 i for all i ≥ 1. In other words, all columns of γ hav e even length and necessarily the length of γ is even. A partition γ is identified with its Y oung diagr am D ( γ ) which is a left and top justified collection of b o xes (or cells) with γ k man y b o xes in the k th row for all k ∈ Z > 0. In particular, the empty Y oung diagram and the partition (0) are identified. The num b er of cells of D ( γ ) is the sum of the parts of γ and is denoted b y | γ | . The b o xes or cells of the Y oung diagram of γ are identified b y its co ordinates ( i, j ) in the matrix st yle, that is, 1 ≤ i ≤  ( γ ) and 1 ≤ j ≤ γ i . Let γ , µ partitions with µ ⊂ γ , that is, µ i ≤ γ i for all i ∈ Z > 0, or the Y oung diagram of µ is a subset of the Y oung diagram of γ . A table au T of (skew) shap e γ /µ is a map (or a filling of D ( γ )) T : D ( γ ) → Z ≥ 0 , T ( i, j ) 7→ T ( i, j ) , THE RECORDING T ABLEAUX IN THE QUANTUM LITTLEWOOD-RICHARDSON MAP 5 assigning a p ositiv e integer to each b o x of γ µ and 0 to the b o xes corresp onding to µ . Denote by T ab( γ /µ ) the set of tableaux of shap e γ /µ . W e say that the tableau T is semi-standar d if an addition the assignment is such that it is weakly increasing as we go from left to right along a row and strictly increasing as w e go from top to b ottom along a column excluding the b o xes in µ , T ( i, j ) ≤ T ( i, j + 1) , T ( i, j ) < T ( i + 1 , j ) , for all ( i, j ) ∈ D ( γ ) \ D ( µ ), and T ( i, j ) = 0 , for ( i, j ) ∈ D ( µ ), where we set T ( a, b ) := ∞ if ( a, b ) / ∈ D ( γ ). Usually T ( i, j ) is just referred as the en try in the b o x ( i, j ) and w e omit the zero es in the b o xes of µ . A p ositiv e integer m ≥  ( γ ) will b e fixed and [0 , m ] := { 0 , 1 , . . . , m } will b e used as a co-domain for map T . W e call [ m ] := { 1 , . . . , m } the alphab et of the semistandard tableau T . In this case, we will denote the set of semistandar d table aux of shap e γ /µ by S S T k ( γ /µ ). When µ = (0), w e just write S S T m ( γ ). The weight or c ontent of T is the nonnegative vector wt( T ) = ( T [1] , . . . , T [ m ]), where T [ i ] := # { ( a, b ) ∈ D ( γ ) : T ( a, b ) = i } for i ∈ [ m ], that is, T [ i ] is the num b er of o ccurrences of i in the tableau T . The r everse r ow wor d of a semi-standard tableau T , denoted w ( T ) = w 1 · · · w l , with l the num b er of non zero entries of T , is obtained by reading the entries of its ro ws (excluding the en try 0) right to left starting from the top row and pro ceeding down wards. The r everse c olumn wor d of a semi-standard tableau T , denoted w col ( T ) = w ′ 1 · · · w ′ l , with l the num b er of non zero entries of T , is obtained b y reading the en tries of its columns (excluding the entry 0) right to left starting from the top and pro ceeding down wards. The weight of the wor d w ( T ) is the weigh t of T . A Y amanouchi wor d is a word u 1 · · · u l suc h that, for each 1 ≤ k ≤ l , the w eight of the subw ord u 1 · · · u k is a partition. 2.1. Sc hensted column insertion and reverse. The c olumn insertion or c olumn bumping [F ul97, Sta98], tak es a p ositive integer x and a tableau T ∈ S S T ( γ ) and puts x in a new b o x at the b ottom of the first column if it is strictly larger than all the entries of the column. If not, it bumps the smal lest entry in the c olumn that is lar ger than or e qual to x . The bump ed entry mov es to the next column, going to the end if p ossible, and bumping an element to the next column otherwise. The pro cess terminates when the bump ed en try go es to the b ottom of the next column, or until it b ecomes the only entry of a new column. The returned tableau is denoted x → T . The r everse c olumn insertion takes a tableau U ∈ S S T ( γ ) and an en try y at the b ottom of a column and bumps it while deletes its b o x. If the column is the first, the new tableau is obtained by deleting the b ottom b o x of the first column of U . If not, the bump ed y mov es to the next column on its left and bumps the lar gest entry in the c olumn that is smal ler than or e qual to y . If the bump ed entry b elongs to the first column the pro cess ends and returns the bump ed en try y ′ from the first column of U together with a new tableau denoted y ← U , with one b o x less, ( y ′ , y ← U ). If not, the pro cess contin ues until an en try is bump ed from the first column of U . Remark 1. [F ul97, App endix A.2, Exercise 3] Let us consider tw o successive column-insertions of x < x ′ in to a tableau T . First column-inserting x in T and then column-inserting x ′ in the resulting tableau x → T , yielding to tw o bumping routes R and R ′ , and tw o new b o xes B and B ′ . Then R ′ lies strictly b elo w R , and B ′ is Southw est of B . If x ≥ x ′ R ′ lies weakly ab o ve R and B ′ Northeast of B . Since column insertion is reversible, one has the following route prop ert y for reverse column insertion. Let y and y ′ b e tw o b ottom column entries of a tableau T , with y ′ South west of y , or let y < y ′ b e the t wo b ottom entries of a column of a tableau T . First consider the reverse column-inserting of y ′ and then rev erse column-inserting y in the resulting tableau y ′ ← T , yielding to tw o bumping routes Z ′ resp ectiv ely Z , and t wo new entries b ′ resp ectiv ely b in the first column of T , T ( i ′ , 1) = b ′ and T ( i, 1) = b . Then i ′ > i and b ′ > b . F or λ ⊆ γ partitions, we write λ ⊂ v ert γ to mean that γ /λ is a vertic al strip . The length of γ /λ is | γ / | λ | := γ | − | λ | . Let λ ∈ P ar ≤ m and k ∈ [0 , m ]. The following assignment [W at25b], combinatorial Pieri’s rule, defined b y the column insertion of a column tableau into a tableau is a bijection 6 OLGA AZENHAS • : S S T m (  k ) × S S T m ( λ ) → G γ ∈ P ar ≤ m λ ⊂ ver t γ , | γ /λ | = k S S T m ( γ ) ( S, T ) 7→ S • T := w k → ( · · · ( w 2 → ( w 1 → T )) · · · ) (9) where S = ( w 1 < w 2 < · · · < w k ) is a column of length k with entries in [1 , k ]. In particular, if S ( i, 1) ≤ T ( i, 1) for every 1 ≤ i ≤  ( T ), and k ≥  ( T ) then S • T = S.T the concatenation of S and T . The pro cess S • T = w k → ( · · · ( w 2 → ( w 1 → T )) is reversible ( S, T ) = w ′ 1 ← ( · · · ( w ′ k − 1 ← ( w ′ k ← S • T )) · · · ) where the en tries of the vertical γ /λ from b ottom to top define the w ord w ′ k · · · w ′ 1 . F or U ∈ S S T m ( γ ) suc h that λ ⊆ v ert γ the inv erse is defined by the reverse column insertion by taking the en tries of U in the v ertical strip λ/γ from b ottom to top. 3. The linear time bijection between the sets e Rec 2 n ( λ/µ ) and LR-Sundaram t ablea ux LRS 2 n ( λ/µ ) Recall partition ν is said to b e even if all columns of ν hav e even length and necessarily the length of ν is even. In this case, ν t the transp ose or conjugate of ν has all rows of even length. W e start by recalling the definition of Littlewoo d-Ric hardson-Sundaram tableau or symplectic Littlewoo d-Richardson tableau. Definition 1. Let λ ∈ P ar ≤ 2 n and µ ∈ P ar ≤ n . A semistandard tableau T ∈ T ab ( λ/µ ) is said to b e a symplectic Littlewoo d-Ric hardson tableau (or Littlew o od-Richardson-Sundaram tableau) if it satisfies the follo wing. (1) T ∈ S S T 2 n ( λ/µ ). (2) Let ( w 1 , . . . , w N ) denote the column-word w col ( T ) of T . Then, the reversed column w ord ( w N , . . . , w 1 ) is a Y amanouc hi w ord. (3) The sequence wt( T ) = ( T [1] , T [2] , . . . , T [2 n ]) is an even partition, (4) If T ( i, j ) = 2 k + 1 for some ( i, j ) ∈ D ( λ/µ ) and k ∈ Z ≥ 0 then we hav e i ≤ n + k . F or λ ∈ P ar ≤ 2 n and µ ∈ P ar ≤ n , the subset of S S T 2 n ( λ/µ ) satisfying conditions (2), (3) and (4) in Definition 1 is denoted by LRS 2 n ( λ/µ ). Remark 2. Let T ∈ LR S 2 n ( λ/µ ) of conten t ν . Put µ (0) := λ and µ ν 1 := µ . F or k = 1 , . . . , ν 1 , define µ ( k ) the shap e obtained by erasing in row i of T restricted to the shap e µ ( k − 1) , the rightmost cell filled with i if an y . Hence µ (0) = λ ⊇ v ert µ (1) ⊇ v ert · · · ⊇ v ert µ ( ν 1 ) = µ This decomp oses the shap e λ/µ into ν 1 v ertical strips or strings J 1 := µ (0) /µ (1) , J 2 := µ (1) /µ (2) , J i = µ ( i − 1) /µ i , . . . , J ν 1 = µ ( ν 1 − 1) /µ ν 1 Let ν t = ( ν t 1 , . . . , ν t ν 1 ) b e the transp ose of ν . Then, for k = 1 , . . . , ν 1 , the vertical strip J k = µ ( k − 1) /µ ( k ) has length ν t k and is filled in T , top to b ottom, with 12 · · · ν t k . Concatenating these string words 12 · · · ν t ν 1 . · · · . 12 · · · ν t 2 . 12 · · · ν t 1 giv es the reverse-column word of the Y amanouchi tableau of shap e ν whic h is also obtained by rectification of the word of T . F or an illustration, see Example 1. Let λ ∈ P ar ≤ 2 n and µ ∈ P ar ≤ n b e such that µ ⊂ λ . A tableau Q of shap e λ/µ is said to b e a r e c or ding table au if there exists T ∈ S S T 2 n ( λ ) such that Q AI I ( T ) = Q . Let R ec 2 n ( λ/µ ) denote the set of r e c or ding table aux of shap e λ/µ . Definition 2. [W at25b]Let λ ∈ P ar ≤ 2 n and µ ∈ P ar ≤ n b e such that µ ⊂ λ . Let e Rec 2 n ( λ/µ ) denote the set of tableaux Q of shap e λ/µ satisfying the following: ( R 1) The entries of Q strictly decrease along the rows from left to right. ( R 2) The entries of Q w eakly decrease along the columns from top to b ottom. ( R 3) F or each k > 0, the num b er Q [ k ] of o ccurrences of k is ev en. THE RECORDING T ABLEAUX IN THE QUANTUM LITTLEWOOD-RICHARDSON MAP 7 ( R 4) F or each k > 0, it holds that Q [ k ] ≥ 2(  ( µ ( k − 1) ) − n ) , where µ ( k − 1) is the partition such that D ( µ ( k − 1) ) = D ( µ ) ∪ { ( i, j ) ∈ D ( λ/µ ) | Q ( i, j ) ≥ k } . ( R 5) F or each r, k > 0, let Q ≤ r [ k ] denote the num b er of o ccurrences of k in Q in the r -th row or ab ov e. Then, the following inequality holds: Q ≤ r [ k + 1] ≤ Q ≤ r [ k ] . Remark 3. Since Q [ k ] ∈ 2 Z and Q [ k + 1] ≤ Q [ k ] ≤  ( λ ) ≤ 2 n , for any k > 0, the weigh t of Q is defined to be the partition wt( Q ) = ( Q [1] , . . . , Q [ λ 1 ]). Its transp ose or conjugate is an even partition ν = ( ν 1 , . . . , ν Q ([1] ). Th us Q is also defined by the sequence of nested partitions µ (0) = λ ⊇ v ert µ (1) ⊇ v ert · · · ⊇ v ert µ ( ν 1 ) = µ where µ i − 1 /µ i are vertical strips of even length Q [ i ] satisfying ( R 4) and ( R 5) conditions, for i = 1 , . . . , ν 1 . As b efore these vertical strips are denoted by J i := µ i − 1 /µ i to emphasize that as vertical strips of Q they are filled with i ’s in Q ., i = 1 , . . . , ν 1 . Prop osition 1. Given Q ∈ e Rec 2 n ( λ/µ ) and k > 0, the following conditions hold ( a ) Q [ k ] = Q ≤ ℓ ( µ ( k − 1) ) [ k ]. ( b ) If 1 ≤ i ≤  ( µ ( k − 1) ) then Q [ k ] = Q ≤ ℓ ( µ ( k − 1) ) [ k ] ≤ Q ≤ i [ k ] + (  ( µ ( k − 1) ) − i ). ( c ) Q [ k ] ≥ 2(  ( µ ( k − 1) ) − n ) (condition ( R 4)) if and only if Q ≤ i [ k ] ≥ 2( i − n ), for all 1 ≤ i ≤  ( µ ( k − 1) ). Pr o of. ( a ) Evident b ecause J k := µ ( k − 1) /µ ( k ) = { ( i, j ) | Q ( i, j ) = k } ⊆ µ ( k − 1) and  ( J k ) = Q [ k ]. Therefore Q ( i, j ) = k ⇒ 1 ≤ i ≤  ( µ ( k − 1) ) . ( b ) Let 1 ≤ i ≤  ( µ ( k − 1) ). Then J k := µ ( k − 1) /µ ( k ) = { ( s, j ) | Q ( s, j ) = k , s ≤ i } ∪ { ( s, j ) | Q ( s, j ) = k , s > i } ⊆ { ( s, j ) | Q ( s, j ) = k , s ≤ i } ∪ { i + 1 , . . . ,  ( µ ( k − 1) ) } Therefore  ( J k ) = Q [ k ] ≤ Q ≤ i [ k ] + (  ( µ ( k − 1) − i ) . ( c ) 1 ≤ i ≤  ( µ ( k − 1) ). F rom ( R 4), Q [ k ] ≥ 2( µ ( k − 1) − n ). By contradiction supp ose, Q ≤ i [ k ] < 2( i − n ). Then Q [ k ] ≤ Q ≤ i [ k ] + (  ( µ ( k − 1) − i ) < 2( i − n ) + (  ( µ ( k − 1) − i ) =  ( µ ( k − 1) − 2 n + i ≤  ( µ ( k − 1) − 2 n +  ( µ ( k − 1) = 2(  ( µ ( k − 1) − n ) . Hence Q [ k ] < 2(  ( µ ( k − 1) − n ) a contradiction with ( R 4). □ Corollary 1. ( a ) F or λ = µ ∈ P ar ≤ n , e Rec 2 n ( λ/λ ) = { D ( λ ) } . ( b ) e Rec 2 n (  l )  = ∅ if and only if l even and l ≤ 2 n . In this case, e Rec 2 n (  l ) = { 1 1 . . . 1 1 } Pr o of. F rom ( R 3), l = Q [1] is even and from ( R 4), Q [1] = l ≥ 2( l − n ) ⇔ l ≤ 2 n . □ 8 OLGA AZENHAS Let ν b e an even partition. Then the ν 1 columns are of even length and its transp ose or conjugate partition ν t = ( ν t 1 , . . . , ν t ν 1 ) is such that ν t i ∈ 2 Z , for all i ≥ 1. The following theorem asserts that the conditions on a LR-Sundaram tableau T translate to conditions on the lengths of their strings J i and gives the right inv erse of the map in [W at25b, Lemma 8.3.2]. Theorem 1. Let T ∈ LRS 2 n ( λ/µ ) of weigh t ν and J 1 , . . . , J ν 1 its decomp osition into vertical strips each strip filled in 1 , 2 , . . . ,  ( J i ) = ν t i ∈ 2 Z for i = 1 , . . . , ν 1 . Relab el each v ertical strip J i = µ ( i − 1) /µ ( i ) with ν t i i ’s, for i = 1 , . . . , ν 1 . W e get a new tableau Q ∈ e Rec 2 n ( λ/µ ) of weigh t ν t . W e denote this map by ♢ . Pr o of. W e hav e to prov e that Q satisfies ( R 4), Q [ r ] ≥ 2(  ( µ ( r − 1) ) − n ) , for each r = 1 , . . . , ν 1 . Let r > 0, n < i ≤  ( µ ( r − 1) ) and ( i, j ) a cell in the vertical strip J r suc h that T ( i, j ) = 2 k + 1, for some k > 0. Since T is an LR-Sundaram tableau it implies n + k ≥ i . One then has Q ( i, j ) = r and Q ≤ i [ r ] = 2 k + 1 and necessarily Q ≤ i [ r ] = 2 k + 1 > 2( i − n ) ⇔ Q ≤ i [ r ] = 2 k + 1 ≥ 2( i − n ) + 1 Otherwise one obtains 2 k + 1 ≤ 2( i − n ) ⇔ 2 k + 2 n ≤ 2 i − 1 ⇔ k + n ≤ i − 1 / 2 ⇒ k + n < i (10) A contradiction with the Sundaram condition. If T ( i + 1 , j ′ ) = 2( k + 1) for some j ′ ≤ j and the cell ( i + 1 , j ′ ) ∈ J r then Q ( i + 1 , j ′ ) = r and Q ≤ i +1 [ r ] = 2( k + 1) = 2 k + 2 ≥ 2( i − n ) + 1 + 1 = 2( i + 1 − n ) By induction on ν 1 . If ν 1 = 1, then ν = (1 m , 0 2 n − m ) with m even. Then T is the vertical strip λ/µ filled with { 1 , . . . , m and Q [1] = m . If  ( λ ) − n ≤ 0 there is nothing to prov e, Q [1] = m > 2(  ( λ ) − n ). Otherwise w e ha ve to c heck the  ( λ ) − n en tries T ( n + 1 , 1) = m − (  ( λ ) − n ) + 1 , . . . , T (  ( λ ) , 1) = m . If  ( λ ) − n is even the entries form a sequence of ℓ ( λ ) − n 2 pairs of o dd and even pairs in consecutive rows in the same column and this case has b een already studied ab o ve. In particular Q [1] = Q ≤ ℓ ( λ ) [1] = m > 2(  ( λ ) − n ) If  ( λ ) − n is o dd we hav e just to chec k T ( n + 1 , 1) = ev en ≤ m b ecause the remaining  ( λ ) − n − 1 en tries w ere already studied ab o ve. In particular Q [1] = Q ≤ ℓ ( λ ) [1] = m > 2(  ( λ ) − n ). Indeed, Q ≤ n +1 [1] = T ( n + 1 , 1) = ev en ≥ 2( n + 1 − m ) = 2 F or ν 1 > 1, let T (1) b e the LRS tableau of shape µ (1) obtained from T by suppressing the string J 1 . Then T 1 ∈ LRS ( µ 1 /µ ), of w eight the even partition ν (1) = ν − (1 ℓ ( ν ) , 0 2 n − ℓ ( ν ) ), that is, its columns are still even and its conjugate partition is ( ν ′ 2 , . . . , ν ′ ν 1 ) and with strings J 2 , . . . J ν 1 . By induction one has Q [ r ] ≥ 2(  ( µ ( r − 1) ) − n ) , for eac h r = 2 , . . . , ν 1 and for each r = 2 , . . . , ν 1 Q ≤ i [ r ] ≥ 2( i − n ) , for 1 ≤ i ≤  ( µ ( r − 1) ) On the other hand b ecause T is an LR tableau Q ≤ r [ k + 1] ≤ Q ≤ r [ k ] , for r ≥ 1 , k > 0 . W e wan t to show that Q [1] ≥ 2(  ( µ (0) ) − n ) = 2(  ( λ ) − n ) One has  ( λ ) ≥  ( µ (1) ) and µ (1) ⊆ λ , henceforth Q ≤ ℓ ( µ (1) ) [1] ≥ Q ≤ ℓ ( µ (1) ) [2] ≥ 2(  ( µ (1) ) − n ) If  ( λ ) =  ( µ (1) ), it is done. Otherwise, λ = ( µ (1) , 1 ℓ ( λ ) − ℓ ( µ (1) ) ) (we are omitting the zero entries in λ and µ (1) ). It remains to chec k the m =  ( λ ) −  ( µ (1) )( ⇔  ( λ ) =  ( µ (1) ) + m ) en tries of T , T (  ( µ (1) ) + 1 , 1) = Q ≤ ℓ ( µ (1) ) [1] + 1 , . . . , T (  ( λ ) , 1) = Q ≤ ℓ ( µ (1) ) [1] + m = Q [1]. As in the case ν 1 = 1 it remains to study the case m = odd ⇔ T (  ( µ (1) ) + 1 , 1) = Q ≤ ℓ ( µ (1) ) [1] + 1 = ev en ⇔ Q ≤ ℓ ( µ (1) ) [1] = odd . If Q ≤ ℓ ( µ (1) ) [1] = odd , then THE RECORDING T ABLEAUX IN THE QUANTUM LITTLEWOOD-RICHARDSON MAP 9 Q ≤ ℓ ( µ (1) ) [1] = odd ≥ Q ≤ ℓ ( µ (1) ) [2] ≥ 2(  ( µ (1) ) − n ) = ev en ⇒ Q ≤ ℓ ( µ (1) ) [1] + 1 ≥ 2(  ( µ (1) ) + 1 − n ) Hence Q ≤ ℓ ( µ (1) ) [1] ≥ 2(  ( µ (1) ) − n ) + 1 ⇒ Q ≤ ℓ ( µ (1) ) [1] + 1 ≥ 2(  ( µ (1) ) − n ) + 2 = 2((  ( µ (1) ) + 1 − n ) Finally , since the cell (  ( µ (1) ) + 1 , 1) ∈ J 1 it means that Q (  ( µ (1) ) + 1 , 1) = 1 and Q ≤ ℓ ( µ (1) )+1 [1] = Q ≤ ℓ ( µ (1) ) [1] + 1 ≥ 2((  ( µ (1) ) + 1 − n ) as desired. (W e hav e assumed that n ≤  ( µ (1) ) which is the worst case.) □ Example 1. Let n = 3, λ = (4 , 3 , 2 , 2 , 1 , 0), µ = (3 , 1 , 0), ν = (3 , 3 , 1 , 1 , 0 2 ) and T ∈ LR S ( λ/µ, ν ) Q = 1 2 1 3 2 3 1 1 ∈ e Rec 6 ( λ/µ ) ♢ ← − ∼ T = 1 1 2 1 2 2 3 4 ∈ LRS 6 ( λ/µ ) (11) with wt( Q ) = (4 , 2 , 2 , 0 3 ) = ν t and the conjugate partition is the even partition ν = (3 , 3 , 1 , 1 , 0 2 ). Q and T are b oth defined b y the sequence of nested partitions µ (0) = λ ⊇ µ (1) = (3 , 2 , 2 , 1 , 0 2 ) ⊇ µ (2) = (3 , 1 , 1 , 0 3 ) ⊇ µ (3) = µ This decomp oses λ/µ in to vertical strips J 1 := µ (0) /µ (1) , J 2 := µ (1) /µ (2) , J 3 = µ (2) /µ The length of the v ertical strip J k = µ ( k − 1) /µ ( k ) is Q [ k ],  ( J k ) =  ( µ ( k − 1) /µ ( k ) ) = Q [ k ], k = 1 , 2 , 3. The vertical strip J i in T is filled with the word 12 . . . ν t i = Q [ i ] and in Q with ν t i = Q [ i ] i ’s. One has Q [ k ] ≥ Q [ k + 1], condition R 5 is verified Q ≤ r [ k + 1] ≤ Q ≤ r [ k ] , for r, k > 0 . as well as condition R 4 is verified. The filling of J 3 , J 2 and J 1 in T are resp ectiv ely the column words 12,12 and 12345. F urthermore concatenating the reverse column words of the strings J 3 , J 2 , J 1 in T giv es the Y amanouchi tableau 1 1 1 2 2 2 3 4 the rectification of the word of T . 3.1. The restriction of the LR orthogonal transp ose linear time map to LR-Sundaram tableaux. In [ACM25] a complete account has b een provided on the LR-symmetries under the action of the dihedral group Z 2 × D 3 . In particular, the LR-symmetry maps are exhibited on Littlew o od-Richardson tableaux as w ell as on the companion pairs. The LR symmetry maps restricted to LR-Sundaram tableaux do not need to return LR-Sundaram tableaux for the same n but its characterization play an imp ortan t role on branching rules from Gl 2 n ( C ) to S 2 n ( C ). Such restriction, in the case of the fundamental symmetry map, c µ,ν,λ = c ν,µ,λ , has b een studied by Kumar-T orres in [KT25] via hiv es which in turn reduces to the restriction of right and left companions as discussed in [Aze25, Aze26]. The restriction of the orthogonal-transp ose linear map ♦ [A CM09, ACM25] exhibiting c µ,ν,λ = c λ t ,ν t ,µ t for LR tableaux, to LR-Sundaram tableaux has not b een studied b efore but surfaces in the branching rule defined by the quantum Littlewoo d-Ric hardson map. It turns out that the R ec 2 n ( λ/µ ) = e Rec 2 n ( λ/µ characterizes explicitly the restriuction of the LR orthogonal transp ose symmetry map on LR-Sundaram tableaux. F or conv enience let us consider our partitions µ, ν, λ inside of a fixed rectangle and define ˇ λ the complemen t of the Y oung diagram of λ inside the given rectangle. With this conv en tion LRS 2 n ( µ, ν, λ ) denotes the set of LR-Sundaram tableaux of shap e λ ∨ /µ and weigh t ν and similarly for e Rec 2 n ( µ, ν t , λ ). See Example 2. 10 OLGA AZENHAS Denote the bijection in Theorem 1 by ♢ . Then the LR orthogonal transp ose symmetry map ♦ [ACM25] is defined by ♦ : LRS 2 n ( µ, ν, λ ) ♢ − → e Rec 2 n ( µ, ν t , λ )  → π ◦ t LR ( λ t , ν t , µ t ) T 7→ Q 7→ Q π ◦ t = ♦ T where Q π ◦ t means the π -rotation (transp osition) follow ed with transp osition (rotation) of D ( λ ). The extra condition ( R 3), in Definition 2, on Q characterizes the LR tableau ♦ T , the image of the LRS tableau T b y the orthogonal transp ose map ♦ . The bijection ♢ and its in verse is so natural that we can intert wine LR-Sundaram tableaux with their Q presentations. F or notation and further details we refer the reader to [Aze99, ACM09, ACM25]. W e resume to Example 1. Example 2. Let n = 3, λ ∨ = (4 , 3 , 2 , 2 , 1), λ = (3 , 2 , 2 , 1) µ = (3 , 1 , 0), ν t = (3 , 3 , 1 , 1 , 0 2 ) and T ∈ LRS ( µ, ν, λ ) T = 1 1 2 1 2 2 3 4 ∈ LRS 6 ( µ, ν, λ ) 7→ Q = 1 2 1 3 2 3 1 1 ∈ e Rec 6 ( µ, ν t , λ ) 7→ Q π ◦ t = 1 1 1 2 2 1 3 3 = ♦ T ♦ T ∈ LR ( λ t , ν t , µ t ) . 4. The cont ainment of the set e Rec 2 n ( λ/µ ) into the set of recording t ableaux Rec 2 n ( λ/µ ) 4.1. Symplectic t ableaux. F rom now on we fix n ∈ N . Definition 3. [Kin76] A semistandard tableau G ∈ S S T 2 n ( γ ) is said to b e symplectic if G ( k , 1) ≥ 2 k − 1 , for all k ∈ [  ( λ )] . Let S pT 2 n ( λ ) denote the set of all symplectic tableaux of shap e γ on the alphab et [2 n ]. Prop osition 2. [W at25b] Let G ∈ S S T 2 n ( λ ). (1) If S pT 2 n ( λ )  = ∅ then  ( λ ) ≤ n . (2) If G is not symplectic, then there exists a unique i ∈ [2 , 2 n ] such that G ( i, 1) < 2 i − 1 and G ( k , 1) ≥ 2 k − 1 for all k ∈ [1 , i − 1] . (12) Moreo ver, w e hav e G ( i − 1 , 1) = 2 i − 3 = 2( i − 1) − 1 and G ( i, 1) = 2 i − 2 = 2( i − 1) . (13) The first symplectic fail needs to b e chec ked in the first column of G among consecutive integer entries consisting of an o dd num b er follow ed with an even num b er, and if the fail o ccurs it happ ens for the first time in a such ev en entry . In other w ords, if the first column has no consecutive integers consisting of an o dd num b er follow ed with an even num b er it is symplectic. Corollary 2. (a) F or 0 ≤ l ≤ n , S p 2 n (  l )  = ∅ . Namely , S p 2 n (()) = { () } , and, for 1 ≤ l ≤ n , G = (1 , 3 , . . . 2 i − 1 , . . . , 2 l − 1) ∈ S p 2 n (  l ) or H = (2 , 4 , . . . , 2 i, . . . , 2 l ) ∈ S p 2 n (  l ). In particular, S p 2 (  1 ) = S S T 2 n (  1 ). (b) S pT 2 n ( λ )  = ∅ if and only if  ( λ ) ≤ n . (c) If G ′ is obtained from G ∈ S p 2 n (  l ) by suppressing 0 ≤ l 0 ≤ l entries then G ′ ∈ S p 2 n (  l − l 0 ). Pr o of. ( a ) The ith entry of G is 2 i − 1, and the i th entry of H is 2 i > 2 i − 1, for i = 1 , . . . , l . ( b ) Consequence of ( a ). Let  ( λ ) = l and let T with first column G or H and the remain columns of T added according to the semistandard-ness and with entries not exceeding 2 n . Then T ∈ S p 2 n ( T ). □ Example 3. Let G ∈ S S T 10 (1 6 ) b e the column (1 , 3 , 4 , 6 , 8 , 9). F or the first symplectic fail we consider the pair 3 , 4 where 3 = 2 × (3 − 1) − 1 and symplectic fail o ccurs in the en try 4 = 2(3 − 1). Removing the entries 3 , 4 it remains the symplectic column (1 , 6 , 8 , 9) ∈ S p 10 (1 6 ). THE RECORDING T ABLEAUX IN THE QUANTUM LITTLEWOOD-RICHARDSON MAP 11 Let G ∈ S S T 6 (1 6 ) with column (1 , 2 , 4 , 5 , 6). The first fail o ccurs in the entry 2 = 2(2 − 1). Removing the entries 1 , 2 it remains the symplectic column (4 , 5 , 6) ∈ S p 6 (1 3 ) Let G ∈ S S T 14 (1 10 ) b e the column (1 , 3 , 4 , 5 , 6 , 7 , 11 , 12 , 13 , 14). The first symplectic fail o ccurs in the en try 4. Removing the pair 3 , 4 it remains (1 , 5 , 6 , 7 , 11 , 12 , 13 , 14) still not symplectic, it fails in the en try 14 = 2(8 − 1) . W e fix λ ∈ P ar ≤ 2 n . The Littlewoo d–Richardson map of type AI I, LR AI I is the algorithm [W at25b, W at25a, Section 3.1] which tak es T ∈ S S T 2 n ( λ ) as input, and returns a pair ( P ( T ) , Q ( T )) of tableaux. Set Rec 2 n ( λ/µ ) := { Q ( T ) | T ∈ S S T 2 n ( λ ) such that sh ( P ( T )) = µ } . (14) An explicit description of the set Rec 2 n ( λ/µ ) is given in [W at25, Theorem 3.1.4 (2)], Rec 2 n ( λ/µ ) = e Rec 2 n ( λ/µ ) b y pro viding a combinatorial pro of for the inclusion Rec 2 n ( λ/µ ) ⊂ e Rec 2 n ( λ/µ ) while the inclusion Rec 2 n ( λ/µ ) ⊃ e Rec 2 n ( λ/µ ) is concluded via representation theory . In turn [W at25b] e Rec 2 n ( λ/µ ) ∼ → LRS 2 n whose combinatorial pro of for the surjectivity we hav e given in Theorem 1 4.2. Reduction map. F or the reader conv enience this section recalls several prop erties of the remov al and successor op erations in [W at25b]. W e fix l ∈ [0 , 2 n ] and a = ( a 1 , . . . , a l ) a column in S S T 2 n (  l ). W e often regard a as a set. The r emoval subwor d of a [W at25b] is defined to b e the subw ord rem( a ) of a obtained by the following recursiv e form ula: rem( a ) :=          ∅ , if l ≤ 1 , rem( a l , . . . , a l − 2 )( a l − 1 , a l ) , if l ≥ 2 , a l ∈ 2 Z , a l − 1 = a l − 1 , and a l < 2 l − | r em ( a 1 , . . . , a l − 2 ) | − 1 , r em ( a 1 , . . . , a l − 1 ) otherwise . (15) Some prop erties of remo v able entries in a . Prop osition 3. [W at25b, Prop osition 2.5.3] (1) If a l ∈ rem ( a ), then a l ∈ 2 Z . (2) If a l / ∈ rem ( a ) then r em ( a ) = r em ( a 1 , . . . , a l − 1 ). (3) If a l is o dd then a l / ∈ rem ( a ), and r em ( a ) = r em ( a 1 , . . . , a l − 1 ). (4) If a l ∈ rem ( a ), then a l ∈ 2 Z and r em ( a 1 , . . . , a l − 1 ) = r em ( a 1 , . . . , a l − 2 ). (5) r em ( a 1 , . . . , a k ) ⊂ rem ( a ) for all k ∈ [0 , l ]. F or each x ∈ Z , set s ( x ) = x + 1, if x / ∈ 2 Z , and s ( x ) = x − 1, if x ∈ 2 Z . Prop osition 4. [W at25b, Prop osition 4.2.2, 4.2.7, Corollary 4.2.8] (1) i ∈ [1 , l ] and a j / ∈ rem ( a ), for j ∈ [ i, l ] then r em ( a ) = rem ( a 1 , . . . , a i − 1 ). (2) for eac h i ∈ [1 , l ], a i ∈ rem ( a ) if and only if one of the following holds (a) a i o dd, i < l , a i +1 = a i + 1 and a i < 2 i − | rem ( a 1 , . . . , a i − 1 ) | (b) a i ev en, i > 1, a i = a i − 1 + 1 and a i < 2 i − | rem ( a 1 , . . . , a i − 2 ) | − 1 (3) a i ∈ r em ( a ) if and only if s ( a i ) ∈ r em ( a ). In particular, if for 1 < i ≤ l , ( a i − 1 , a i ) with a i ev en, is a remov able pair from a , then a i < 2 i − 1 and a i is a symplectic failing entry . Consequen tly , | r em ( a ) | ∈ 2 Z . (4) 0 ≤ l − | r em ( a ) | ≤ min (2 n − l, l ). In particular, | r em ( a ) | ≥ 2( l − n ). (5) F or i ∈ [1 , l ], (a) a i o dd, then | r em ( a 1 , . . . , a i − 1 ) | ≥ 2 i − a i − 1 (b) a i ev en, then | r em ( a 1 , . . . , a i ) | ≥ 2 i − a i The following is a consequence of previous prop osition p oin ts (2) and (3). 12 OLGA AZENHAS Corollary 3. (1) a is symplectic ⇔ r em ( a ) = ∅ . (2) r em ( a ) = ∅ if and only if either l = 1 or l ≥ 2 in which case, for each 1 ≤ i ≤ l , one of the following holds (S1) a i ∈ 2 Z (S2) a i / ∈ 2 Z and a i +1 ≥ a i + 2, for i < l (S3) a i / ∈ 2 Z , a i +1 = a i + 1 and a i ≥ 2 i + 1, i < l Pr o of. Condition ( S 3) ensures if a i / ∈ 2 Z , a i +1 = a i + 1 and a i ≥ 2 i + 1 then 2 Z ∋ a i +1 = a i + 1 ≥ 2 i + 1 + 1 = 2( i + 1) ≥ 2(+1) − 1. If a 1 ∈ 2 Z then a 1 ≥ 2 × − 1. If a i ∈ 2 Z with i > 1 and a i = a i − 1 + 1, ( S 3) ensures a i − 1 ≥ 2( i − 1) + 1 ⇒ a i = a i − 1 + 1 ≥ 2( i − 1) + 2 = 2 i ≥ 2 i − 1. If a k ∈ 2 Z for all 1 ≤ k ≤ i then a i ≥ 2 i . If a 1 ≥ 1 and a k ≥ a k − 1 + 2 for 1 ≤ k ≤ i , then a k ≥ 2 k − 1. If a i / ∈ 2 Z and a i +1 ≥ a i + 2 then a i ≥ 2 i − 1 ⇒ a i +1 ≥ 2 i − 1 + 2 = 2( i + 1) − 1 If a k / ∈ 2 Z for all 1 ≤ k ≤ i then a k ≥ 2 k − 1. If a k − 1 ∈ 2 Z and a k / ∈ 2 Z , since w e ha ve seen that a k − 1 ≥ 2( k − 1) then a k ≥ a k − 1 + 1 ≥ 2( k − 1) + 1 = k − 1. □ Corollary 4. (1) Let k ∈ [1 , n ]. Then S pT 2 n (  k ) = { S = ( a 1 , . . . , a k ) ∈ S S T 2 n (  k ) |∀ i ∈ [1 , k ] , ( S 1) ∨ ( S 2) ∨ ( S 3) } (2) Let λ ∈ P ar ≤ n , S pT 2 n ( λ ) = { S ∈ S S T 2 n ( λ ) | ( S (1 , 1) , · · · , S (  ( λ ) , 1) ∈ S pT 2 n (  ℓ ( λ ) ) } Example 4. r em (4 , 5 , 6 , 7 , 8) = (7 , 8), r ed (4 , 5 , 6 , 7 , 8) = (4 , 5 , 6) r em (1 , 2 , 3 , 4 , 8) = (1 , 2 , 3 , 4), r ed (1 , 2 , 3 , 4 , 8) = (8). The next corollary follows from the definition of remov al subw ord in (15) and item (2) in the previous prop osition. Corollary 5. F or l ∈ [0 , 2 n ], r em ( a ) = a if and only if l is even and a = (1 , 2 , . . . , l ) Pr o of. F or l = 0, there is nothing to prov e, a = ∅ and r em ( ∅ ) = ∅ . If l = 1, it follo ws from (15), r em ( a ) = ∅ . Let l ≥ 2, and a = ( a 1 , . . . , a l ). Let us prov e that a 1 , a 2 ∈ r em ( a ) ⇔ a 1 = 1 and a 2 = 2. F rom Prop osition 4, (2), a 1 ∈ rem ( a ) ⇔ 1 ≤ a 1 o dd , a 2 = a 1 + 1 and a 1 < 2 − | rem ( ∅ ) | = 2 This implies a 1 = 1 and a 2 = 2 a 2 = 2 ∈ rem ( a ) ⇔ 2 = a 2 = a 1 + 1 = 1 + 1 even and 2 < 2 − | r em ( ∅ ) | − 1 = 4 − 1 = 3 By induction assume a 1 , . . . , a k ∈ r em ( a 1 , . . . , a k , . . . , a l ) ⇔ k even and a i = i , for i = 1 , . . . , k . Then from Prop osition 4, (2), a k +1 ∈ rem ( a ) ⇔ a k +1 o dd, k + 1 < l , a k +2 = a k +1 + 1 and k = a k ev en < a k +1 odd < 2 k + 1 − | r em ( a 1 , . . . , a k ) | = 2( k + 1) − k = k + 2 (Note that a k +1 ev en ⇒ a k +1 = a k + 1 o dd whic h is absurd.) Therefore a k +1 = k + 1 o dd and a k +2 = k + 2 ev en. Also a k +2 = k + 2 ∈ r em ( a ) since a k +2 ev en, k + 2 > 1, k + 2 = a k +2 = a k +1 + 1 and k + 2 = a k +2 < 2( k + 2) − | rem ( a 1 , . . . , a k ) | − 1 = 2( k + 2) − k − 1 = k + 3. □ Definition 4. [W at25b]F or the column a , the new column red( a ) is defined to b e the one obtained from a b y remo ving the en tries in the set rem( a ). Definition 5. [W at25b]Let S ∈ S S T 2 n . Let S 1 denote the first column of S , and S ≥ 2 the rest of the tableau S . The new tableau suc ( S ) is defined to b e suc( S ) := r ed ( S 1 ) • S ≥ 2 the Schensted column insertion of red( S 1 ) in S ≥ 2 . (16) THE RECORDING T ABLEAUX IN THE QUANTUM LITTLEWOOD-RICHARDSON MAP 13 In addition the suc c essor map on S S T 2 n is the assignment S S T 2 n ( λ ) → G µ ∈ P ar ≤ 2 n S S T 2 n ( µ ) (17) S 7→ suc ( S ) = red( S 1 ) • S ≥ 2 (18) In particular, if S is a column suc( S ) = red( S ). F rom [W at25b, Corollary 4.4.4] the reduction and successor m aps are injectiv e. Some prop erties of the reduction map and the remov al set follow. Prop osition 5. [W at25b, NSW26] Let l ∈ [0 , 2 n ] and a = ( a 1 , . . . , a l ) ∈ S S T 2 n (  l ) where  l = (1 l ). Then (1) l even ⇒ r em (1 , 2 , . . . , l ) = (1 , 2 , . . . , l ) = red (1 , 2 , . . . , l ) = ∅ . (2) l o dd ⇒ r em (1 , 2 , . . . , l ) = r em (1 , 2 , . . . , l − 1) = (1 , 2 , . . . , l − 1) ⇒ r ed (1 , 2 , . . . , l ) = { l } . (3) r ed ( a ) is symplectic. Moreov er, the reduction map on S S T 2 n (  l ) is injective, r ed : S S T 2 n (  l ) → G 0 ≤ k ≤ min { l, 2 n − l } l − k ∈ 2 Z S pT 2 n (  k ) a 7→ red ( a ) = suc 1 ( a ) (19) Corollary 6. F or l ∈ [0 , 2 n ], r ed ( a ) = ∅ if and only if l is even and a = (1 , 2 , . . . , l ) If a = ( a 1 , . . . , a l ), r ed ( a ) = a l if and only if l / ∈ 2 Z . Prop osition 6. [W at25b, NSW26] Let T ∈ S S T 2 n ( λ ) and let µ b e the shap e of suc ( T ). Then (a) µ ⊂ λ , λ/µ is a vertical strip. (b) λ = µ if and only if T = suc ( T ). (c) T is symplectic if and only if suc ( T ) = T . (d) The suc map on S S T 2 n ( λ ) is injective. Definition 6. [W at25b]Let T ∈ S S T 2 n ( λ ) and let N be a nonnegative integer satisfying suc N +1 ( T ) = suc N ( T ). Set P AI I ( T ) = suc N ( S ) and Q AI I ( T ) to b e the tableau that records the pro cess of transformations from T to P AI I ( T ). Let the shap es of T , suc 1 ( T ), suc 2 ( T ) , . . . , suc N ( T ) be λ 0 = λ ⊃ λ 1 ⊃ λ 2 ⊃ · · · ⊃ λ N = sh ( P AI I ( T )) resp ectiv ely , and define Q AI I ( T ) as the tableau obtained b y placing the en try j in λ ( j − 1) /λ j for j = 1 , 2 , . . . , N F or λ ∈ P ar ≤ 2 n and µ ∈ P ar ≤ n , set Rec 2 n ( λ/µ ) := { Q AI I ( T ) | T ∈ S S T 2 n ( λ ) such that sh ( P AI I ( T )) = µ } . (20) W atanab e has shown combinatorially in [W at25b] that Rec 2 n ( λ/µ ) ⊂ e Rec 2 n ( λ/µ ) and thus that LR AI I : S S T 2 n ( λ ) − → G µ ∈ P ar ≤ n µ ⊂ λ S pT 2 n ( µ ) × e Rec 2 n ( λ/µ ) T 7→ ( suc N ( T ) , Q ) (21) is an injection. In particular, S pT 2 n ( λ ) ⊂ S S T 2 n ( λ ) and LR AI I ( S ) = (suc( S ) , Q ( S ) = ( S, D ( λ/λ )) = ( S, ∅ ) for all S ∈ S pT 2 n ( λ ). 4.3. Expanding map by reverse Schensted column insertion and reverse remov als. T o prov e that LR AI I is also a surjection we exhibit the righ t inv erse e R of LR AI I . Fix arbitrarily S ∈ S pT 2 n ( µ ) with µ ∈ P ar ≤ n and µ ⊂ λ , and define e R : { S } × e Rec 2 n ( λ/µ ) − → S S T 2 n ( λ ) ( S, Q ) 7→ T = suc − 1 ( S ) (22) suc h that LR AI I ◦ e R ( S, Q ) = ( S, Q ). This can b e done by induction on the n umber of strings of Q b y un winding the successive successors in Definition 6. The fundamental step is the un wind of a pair 14 OLGA AZENHAS ( S, Q ) ∈ S p 2 n ( µ ) × e Rec 2 n (( λ 0 ,  l − ℓ ( µ ) ) /µ ) with µ ⊆ λ 0 ∈ P ar ≤ n suc h that  ( λ 0 ) =  ( µ ) and λ 0 /µ is a v ertical strip. 4.3.1. One string in Q . Case λ =  l . S ∈ S pT 2 n (  k ) and Q ∈ e Rec 2 n (  l / k ) with 0 ≤ k ≤ n and k ≤ l ≤ 2 n . F rom ( R 3) and ( R 4), it follows l − k = Q [1] ∈ 2 Z ≥ 2( l − n ) ⇔ k ≤ 2 n − l , k ≤ l and l − k ∈ 2 Z If l = k ≤ n , e Rec 2 n (  l / l ) = { () } and S symplectic of length l , S ∈ S p 2 n (  l ). e R : { S } × e Rec 2 n (())) − → S S T 2 n (  l ) (23) ( S, Q = ()) 7→ S, r ed ( S ) = S (24) If k = 0 ≤ l ∈ 2 Z , µ = ∅ ⇔ S = ∅ and, for l ≤ 2 n ev en, one has | e Rec 2 n (  l ) | = 1, and e R : {∅} × e Rec 2 n (  l ) − → S S T 2 n (  l ) (25) ( ∅ , Q ) 7→ T = (1 , 2 , . . . , l ) (26) F rom Corollary 1, e Rec 2 n (  l )  = ∅ ⇔ l ∈ 2 Z and l ≤ 2 n, and, in this case, e Rec 2 n (  l ) = { Q = 1 1 . . . 1 1 } , l ∈ 2 Z , and l ≤ 2 n F rom Corollary 6, for l even, T ∈ S S T 2 n (  l ) : r ed ( T ) = ∅ ⇔ T = (1 , . . . , l ) F rom Definition 2, ( R 2) and ( R 3), if 2 ≤ l − k ∈ 2 Z , l − k ≥ 2( l − n ) ⇔ l ≤ 2 n − k , and since S is such that S ∈ S pT 2 n (  k ), 1 ≤ k ≤ n , Q ∈ e Rec 2 n (  l / k ), e Rec 2 n (  l / k ) = { . . . 1 1 . . . 1 1 , k ≤ n, 2 ≤ l − k ∈ 2 Z , l ≤ 2 n − k } Let S ∈ S p (  k ), e R : { S } × e Rec 2 n (  l / k ) − → S S T 2 n (  l ) ( S, Q ) 7→ T = suc − 1 ( S ) (27) W e ha v e to show that the reduction of T , suc ( T ) = r ed ( T ) = S where | r em ( T ) | = l − k p ositiv e even ≥ 2( l − n ). Prop osition 7. Let k = 1, S = ( b ) ∈ S p 2 n (  1 ), Q ∈ e Rec 2 n (  l / 1 ) such that l ≤ 2 n − 1, 2 ≤ l − 1 even. Let l ′ and l ′′ nonnegativ e ev en num b ers such that l ′ + l ” = l − 1 and l ′ = ( b − 2 , if b ∈ 2 Z b − 1 , b / ∈ 2 Z (28) THE RECORDING T ABLEAUX IN THE QUANTUM LITTLEWOOD-RICHARDSON MAP 15 Then e R ( S, Q ) = T = ( a 1 , . . . , a l ) where T = ( (1 , 2 , . . . b − 2 , b, b + 1 , b + 2 , . . . , b + ( l − 1 − ( b − 2))) if b ∈ 2 Z (1 , 2 , . . . b − 1 , b, b + 1 + 1 , b + 1 + 2 , . . . , b + 1 + ( l − 1 − ( b − 1))) , b / ∈ 2 Z (29) Pr o of. W e hav e to show that T ∈ S S T 2 n (  l ) and r ed ( T ) = ( b ). Indeed  ( T ) = l ′ + l ′′ + 1 = l and b + l ” = ( b + l − l ′ − 1 = b + l − b + 2 − 1 = l + 1 ≤ 2 n − 1 + 1 = 2 n if b ∈ 2 Z b + l − l ′ − 1 = b + l − b + 1 − 1 = l − 1 ≤ 2 n − 2 ⇒ b + l ” + 1 ≤ 2 n − 1 , b / ∈ 2 Z (30) Hence T ∈ S S T 2 n (  l ). F rom Prop osition 4, (2), (3) r em ( T ) = ( (1 , 2 , . . . l ′ , b + 1 , b + 2 , . . . , b + l ”) if b ∈ 2 Z (1 , 2 , . . . l ′ , b + 1 + 1 , b + 1 + 2 , . . . , b + 1 + l ”) , b / ∈ 2 Z (31) Let 1 ≤ i ≤ l ”, then                              a l ′ +1+ i = b + i ≤ 2( l ′ + 1 + i ) − | rem (1 , 2 , . . . l ′ , b, b + 1 , . . . , b + i − 1) | if i o dd and b ∈ 2 Z a l ′ +1+ i +1 = b + i + 1 ≤ 2( l ′ + 1 + i + 1) − | rem (1 , 2 , . . . l ′ , b, b + 1 , . . . , b + i − 1) | − 1 if i o dd and b ∈ 2 Z a l ′ +1+ i = b + 1 + i ≤ 2( l ′ + 1 + i ) − | rem (1 , 2 , . . . l ′ , b, b + 1 + 1 , . . . , b + 1 + i − 1) | , if i o dd and b / ∈ 2 Z a l ′ +1+ i +1 = b + 1 + i + 1 ≤ 2( l ′ + 1 + i + 1) − | rem (1 , 2 , . . . l ′ , b, b + 1 + 1 , . . . , b + 1 + i − 1) | − 1 , if i o dd and b / ∈ 2 Z (32) F or b ∈ 2 Z , | r em (1 , 2 , . . . l ′ , b ) | = | r em (1 , 2 , . . . l ′ ) | = l ′ . By induction on i ≥ 1, | r em (1 , 2 , . . . l ′ , b, b + 1 , . . . , b + i − 1) | = l ′ + i − 1 and 2( l ′ + 1 + i ) − | rem (1 , 2 , . . . l ′ , b, b + 1 , . . . , b + i − 1) | = 2 l ′ + 2 + 2 i − l ′ − i + 1 = l ′ + i + 3 = b − 2 + i + 3 = b + i + 1 > b + i for i o dd F or b / ∈ 2 Z , by induction on i ≥ 1, | r em (1 , 2 , . . . l ′ , b, b + 1 + 1 , . . . , b + 1 + i − 1) | = l ′ + i − 1 and 2( l ′ + 1 + i ) − | rem (1 , 2 , . . . l ′ , b, b + 1 , . . . , b + i − 1) | = 2 l ′ + 2 + 2 i − l ′ − i + 1 = l ′ + i + 3 = b − 1 + i + 3 = b + i + 2 > b + i + 1 for i o dd Hence r ed ( T ) = ( b ) □ Observ e that from Prop osition 4, (2),(3) S p 2 n (  2 ) = { S = ( a 1 < a 2 ) ∈ S S T 2 n (  2 ) : a 1 ∈ 2 Z } ⊔ { S = ( a 1 < a 2 ) ∈ S S T 2 n (  2 ) : a 1 / ∈ 2 Z ∧ a 2 ≥ a 1 + 2 } ⊔ { S = ( a 1 < a 2 ) ∈ S S T 2 n (  2 ) : a 1 / ∈ 2 Z ∧ a 2 = a 1 + 1 ∧ a 1 ≥ 3 } (33) Prop osition 8. Let k = 2 ≤ n , S = ( a 1 , a 2 ) ∈ S p 2 n (  2 ), Q ∈ e Rec 2 n (  l / 2 ) suc h that l ≤ 2 n − 2, 2 ≤ l − 2 ev en. Let l 1 , l 2 , l 3 nonnegativ e ev en num b ers such that l 1 + l 2 + l 3 = l − 2 and l 1 = ( a 1 − 2 , if a 1 ∈ 2 Z a 1 − 1 , a 1 / ∈ 2 Z l 2 =      a 2 − a 1 − 2 a 1 ∈ 2 Z , a 2 ∈ 2 Z or a 1 / ∈ 2 Z , a 2 / ∈ 2 Z ∧ a 2 − a 1 ≥ 3 a 2 − a 1 − 1 a 1 ∈ 2 Z , a 2 / ∈ 2 Z or a 1 / ∈ 2 Z , a 2 ∈ 2 Z ∧ a 2 − a 1 ≥ 3 0 , otherwise (34) 16 OLGA AZENHAS Then e R ( S, Q ) = T = T 0 .T 1 .T 2 ∈ S S T 2 n (  l ) where T =                              (1 , 2 , . . . l 1 , a 1 , a 1 + 1 , . . . , a 1 + l 2 , a 2 , a 2 + 1 , . . . , a 2 + l 3 ) if a 1 , a 2 ∈ 2 Z (1 , 2 , . . . l 1 , a 1 , a 1 + 1 , . . . , a 1 + l 2 , a 2 , a 2 + 1 + 1 , . . . , a 2 + 1 + l 3 ) , if a 1 ∈ 2 Z , a 2 / ∈ 2 Z (1 , 2 , . . . l 1 , a 1 , a 1 + 1 + 1 , a 1 + 1 + 2 , . . . , a 1 + 1 + l 2 , a 2 , a 2 + 1 + 1 , . . . , a 2 + 1 + l 3 ) , if a 1 , a 2 / ∈ 2 Z (1 , 2 , . . . l 1 , a 1 , a 1 + 1 + 1 , a 1 + 1 + 2 , . . . , a 1 + 1 + l 2 , a 2 , a 2 + 1 , . . . , a 2 + l 3 ) , if a 1 / ∈ 2 Z , a 2 ∈ 2 Z . (35) Pr o of. W e hav e to show that T ∈ S S T 2 n (  l ) and r ed ( T ) = ( a 1 , a 2 ) a 2 + l 3 =                              a 2 + l − l 1 − l 2 − 2 = a 2 + l − a 1 + 2 − a 2 + a 1 + 2 − 2 = l + 2 ≤ 2 n − 2 + 2 = 2 n if a 1 , a 2 ∈ 2 Z a 2 + l − l 1 − l 2 − 2 = a 2 + l − a 1 + 2 − a 2 + a 1 + 1 − 2 = l + 1 ≤ 2 n − 2 + 1 = 2 n − 1 ⇒ a 2 + l 3 + 1 ≤ 2 n, a 1 ∈ 2 Z , a 2 / ∈ 2 Z a 2 + l − l 1 − l 2 − 2 = a 2 + l − a 1 + 1 − a 2 + a 1 + 2 − 2 = l + 1 ≤ 2 n − 2 + 1 = 2 n − 1 ⇒ a 2 + l 3 + 1 ≤ 2 n, a 1 , a 2 / ∈ 2 Z a 2 + l − l 1 − l 2 − 2 = a 2 + l − a 1 + 1 − a 2 + a 1 + 1 − 2 = l ≤ 2 n − 2 = 2 n − 2 a 1 / ∈ 2 Z , a 2 ∈ 2 Z (36) Hence, from Prop osition 4, (2), (3), T ∈ S S T 2 n (  l ). □ e Rec 2 n (( µ,  l − ℓ ( µ ) ) /µ ) = { . . . 1 1 . . . 1 1 ,  ( µ ) ≤ n, l −  ( µ ) ∈ 2 Z , l ≤ 2 n −  ( µ ) } Prop osition 9. Let µ ∈ P ar ≤ n , S ∈ S p 2 n ( µ ), Q ∈ e Rec 2 n (( µ,  l − ℓ ( µ ) ) /µ ) such that l ≤ 2 n −  ( µ ), 0 ≤ l −  ( µ ) ev en. Let S 1 = ( a 1 , . . . , a ℓ ( µ ) ) ∈ S p 2 n (  ℓ ( µ ) ) b e the first column of S and let be S ≥ 2 the rest part of S with k =  ( µ ), and l 1 , l 2 , . . . , l ℓ ( µ ) , l ℓ ( µ )+1 nonnegativ e ev en num b ers suc h that l 1 + l 2 + · · · + l ℓ ( µ )+1 = l −  ( µ ) and l 1 = ( a 1 − 2 , if a 1 ∈ 2 Z a 1 − 1 , if a 1 / ∈ 2 Z , (37) l i +1 =                a i +1 − a i − 2 , if a i ∈ 2 Z , a i +1 ∈ 2 Z or a i / ∈ 2 Z , a i +1 / ∈ 2 Z ∧ a i +1 − a i ≥ 2 i + 1 a i +1 − a i − 1 , if a i ∈ 2 Z , a i +1 / ∈ 2 Z or a i / ∈ 2 Z , a i +1 ∈ 2 Z ∧ a i +1 − a i ≥ 2 i + 1 0 , otherwise , 1 ≤ i <  ( µ ) . (38) Then e R ( S, Q ) = T = ( T 0 T 1 · · · T ℓ ( µ ) ) .S ≥ 2 ∈ S S T 2 n (( µ,  l − ℓ ( µ ) )) where THE RECORDING T ABLEAUX IN THE QUANTUM LITTLEWOOD-RICHARDSON MAP 17 T 0 = (1 , 2 , . . . l 1 ) , (39) T i = ( ( a i , a i + 1 , . . . , a i + l i +1 ) , if a i ∈ 2 Z ( a i , a i + 1 + 1 , a 1 + 1 + 2 , . . . , a i + 1 + l i +1 ) , if a i / ∈ 2 Z 1 ≤ i ≤  ( µ ) . (40) Pr o of. W e hav e to show that T ∈ S S T 2 n (( µ,  l − ℓ ( µ ) )) and r ed (( T 0 T 1 · · · T l )) = S 1 . Indeed ( T 0 T 1 · · · T ℓ ( µ ) ) the first column of T satisfies ( T 0 T 1 · · · T ℓ ( µ ) ) ∈ S S T 2 n (  l ):  ( T 0 T 1 · · · T l ) = l 1 + l 2 + · · · + l ℓ ( µ )+1 +  ( µ ) = l and T is a semistandard tableau b ecause the entries of S 1 are included in ( T 0 T 1 · · · T l ). More precisely , each a i b elongs to T i for i = 1 , . . . , l , and therefore pushed down in the first column of T . Hence the column insertion of ( T 0 T 1 · · · T l )) in S ≥ 2 is just the concatenation ( T 0 T 1 · · · T ℓ ( µ ) ) .S ≥ 2 . Note l ℓ ( µ )+1 = l − ℓ ( µ ) X i =1 l i −  ( µ ) . Let a 0 := 0 ∈ 2 Z and note that from (37) ℓ ( µ ) X i =1 l i ≥ ℓ ( µ ) X i =1 ( a i − a i − 1 − 2) = a ℓ ( µ ) − 2  ( µ ) . If a ℓ ( µ ) / ∈ 2 Z let a j − 1 , a j , 0 ≤ j ≤  ( µ ) b e the first pair of entries of S 1 suc h that a j − 1 ∈ 2 Z and a j / ∈ 2 Z in which case one has a j − a j − 1 − 1 = l j . Then ℓ ( µ ) X i =1 l i ≥ ℓ ( µ ) X i =1 i  = j ( a i − a i − 1 − 2) + ( a j − a j − 1 − 1) = a ℓ ( µ ) − 2  ( µ ) + 1 . Then a ℓ ( µ ) + l ℓ ( µ )+1 = a ℓ ( µ ) + l − ℓ ( µ ) X i =1 l i −  ( µ ) ≤ a ℓ ( µ ) + l − ( a ℓ ( µ ) − 2  ( µ )) −  ( µ ) = l +  ( µ ) ≤ 2 n, if a ℓ ( µ ) ∈ 2 Z and a ℓ ( µ ) + l ℓ ( µ )+1 + 1 = a ℓ ( µ ) + 1 + l − ℓ ( µ ) X i =1 l i −  ( µ ) ≤ a ℓ ( µ ) + l + 1 − ( a ℓ ( µ ) − 2  ( µ ) + 1) −  ( µ ) = l +  ( µ ) ≤ 2 n, if a ℓ ( µ ) / ∈ 2 Z □ e Rec 2 n (( λ 0 ,  l − ℓ ( µ ) ) /µ ) = { 1 . . . 1 1 1 . . . 1 1 ,  ( µ ) ≤ n, l 0 = | λ 0 | − | µ | l −  ( µ ) + l 0 ∈ 2 Z , l ≤ 2 n −  ( µ ) + l 0 } 18 OLGA AZENHAS Let µ ⊆ λ 0 ∈ P ar ≤ n suc h that  ( λ 0 ) =  ( µ ) and λ 0 /µ is a vertical strip of length l 0 = | λ 0 | − | µ | . Let S ∈ S p 2 n ( µ ) and Q ∈ e Rec 2 n (( λ 0 ,  l − ℓ ( µ ) ) /µ ) such that l ≤ 2 n −  ( µ ) + l 0 ⇔ l +  ( µ ) − l 0 ≤ 2 n, and 0 ≤ l −  ( µ ) + l 0 ∈ 2 Z . (41) Observ e that these conditions are forced by Definition 2, ( R 3), ( R 4) on Q . F or instance, for = 4, Q = 1 1 1 ∈ e Rec 2 n (( λ 0 ,  l − ℓ ( µ ) ) /µ ) , µ = (4 , 2 , 2 , 1) ⊆ λ 0 = (4 , 3 , 2 , 2) ,  ( λ 0 ) =  ( µ ) = 4 , l 0 = 2 , l = 5 Let r 1 < · · · < r ℓ ( µ ) − l 0 b e the complement of the set of the row co ordinates of the cells of the vertical strip λ 0 /µ in [1 ,  ( µ )]. Let S ′ 1 = ( a 1 , . . . , a ℓ ( µ ) − l 0 ) ∈ S p 2 n (  ℓ ( µ ) − l 0 ) b e the set of the bump ed elements from the first column S 1 of S by applying successiv ely the rev erse column insertion to the ro ws r ℓ ( µ ) − l 0 < · · · < r 1 (from the largest to the smallest ro w) of S , and S ′ ≥ 2 the returned tableau b y the application of that reverse column insertion on S . Note that Remark 1 ensures that S ′ 1 is w ell defined and since S is symplectic and S ′ 1 is contained in the first column of S , S ′ 1 is also symplectic. Let l 1 , l 2 , . . . , l ℓ ( µ ) − l 0 , l ℓ ( µ ) − l 0 +1 b e nonnegative ev en num b ers such that ℓ ( µ ) − l 0 X i =1 l i + l ℓ ( µ ) − l 0 +1 = l −  ( µ ) + l 0 ⇔ l ℓ ( µ ) − l 0 +1 = l −  ( µ ) + l 0 − ℓ ( µ ) − l 0 X i =1 l i (42) and l 1 = ( a 1 − 2 , if a 1 ∈ 2 Z a 1 − 1 , if a 1 / ∈ 2 Z , (43) l i +1 =                a i +1 − a i − 2 , if a i ∈ 2 Z , a i +1 ∈ 2 Z or a i / ∈ 2 Z , a i +1 / ∈ 2 Z ∧ a i +1 − a i ≥ 2 i + 1 a i +1 − a i − 1 , if a i ∈ 2 Z , a i +1 / ∈ 2 Z or a i / ∈ 2 Z , a i +1 ∈ 2 Z ∧ a i +1 − a i ≥ 2 i + 1 0 , otherwise , 1 ≤ i <  ( µ ) − l 0 . (44) Theorem 2. With the setting ab o ve, e R ( S, Q ) = T = ( T 0 T 1 · · · T ℓ ( µ ) − l 0 ) .S ′ ≥ 2 ∈ S S T 2 n (( λ 0 ,  l − ℓ ( µ ) )) where T 0 = (1 , 2 , . . . l 1 ) , (45) T i = ( ( a i , a i + 1 , . . . , a i + l i +1 ) , if a i ∈ 2 Z ( a i , a i + 1 + 1 , a 1 + 1 + 2 , . . . , a i + 1 + l i +1 ) , if a i / ∈ 2 Z 1 ≤ i ≤  ( µ ) − l 0 . (46) Pr o of. F rom F rom Prop osition 4, (2), (3), r em ( T 0 T 1 · · · T ℓ ( µ ) − l 0 ) = T 0 T 1 · · · T ℓ ( µ ) − l 0 \ S ′ 1 ⇒ red ( T 0 T 1 · · · T ℓ ( µ ) − l 0 ) = S ′ 1 . W e ha ve to show that ( T 0 T 1 · · · T ℓ ( µ ) − l 0 ) .S ′ ≥ 2 ∈ S S T 2 n (( λ 0 ,  l − ℓ ( µ ) )). By construction the shap e is ( λ 0 ,  l − ℓ ( µ ) ). Indeed ( T 0 T 1 · · · T ℓ ( µ ) − l 0 ), the first column of T , satisfies ( T 0 T 1 · · · T ℓ ( µ ) − l 0 ) ∈ S S T 2 n (  l ). Its THE RECORDING T ABLEAUX IN THE QUANTUM LITTLEWOOD-RICHARDSON MAP 19 length is  ( T 0 T 1 · · · T ℓ ( µ ) − l 0 ) = ℓ ( µ ) − l 0 X i =1 l i + l ℓ ( µ ) − l 0 +1 +  ( S ′ 1 ) = ℓ ( µ ) − l 0 X i =1 l i + l ℓ ( µ ) − l 0 +1 +  ( µ ) − l 0 ,  ( S ′ 1 ) =  ( µ ) − l 0 = l by (42). and its entries are b ounded by 2 n as we show next. Note that from (42) l ℓ ( µ ) − l 0 +1 = l −  ( µ ) + l 0 − ℓ ( µ ) − l 0 X i =1 l i . Let a 0 := 0 ∈ 2 Z and note that from (43), (44), ℓ ( µ ) − l 0 X i =1 l i ≥ ℓ ( µ ) − l 0 X i =1 ( a i − a i − 1 − 2) = a ℓ ( µ ) − l 0 − 2(  ( µ ) − l 0 ) . (47) If a ℓ ( µ ) − l 0 / ∈ 2 Z , we may refine the previous inequality . Either all a i , for i = 1 , . . . ,  ( µ ) − l 0 , are o dd in whic h case l 1 = a 1 − 1 from (43), or let a j − 1 , a j , 0 ≤ j ≤  ( µ ) − l 0 b e the first pair of entries of S ′ 1 suc h that a j − 1 ∈ 2 Z and a j / ∈ 2 Z (this includes the previous case with a 0 = 0) in which case one has, from (43), (44), a j − a j − 1 − 1 = l j . Then ℓ ( µ ) − l 0 X i =1 l i ≥ ℓ ( µ ) − l 0 X i =1 i  = j ( a i − a i − 1 − 2) + ( a j − a j − 1 − 1) = a ℓ ( µ ) − l 0 − 2(  ( µ ) − l 0 ) + 1 . (48) Henceforth, from (42) and (47), (48) a ℓ ( µ ) − l 0 + l ℓ ( µ ) − l 0 +1 =                                            a ℓ ( µ ) − l 0 + l −  ( µ ) + l 0 − P ℓ ( µ ) − l 0 i =1 l i ≤ a ℓ ( µ ) − l 0 + l −  ( µ ) + l 0 − ( a ℓ ( µ ) − l 0 − 2(  ( µ ) − l 0 )) , by (47) = l +  ( µ ) − l 0 ≤ 2 n, b y (42) , if a ℓ ( µ ) − l 0 ∈ 2 Z and a ℓ ( µ ) − l 0 + l −  ( µ ) + l 0 − P ℓ ( µ ) − l 0 i =1 l i ≤ a ℓ ( µ ) − l 0 + l −  ( µ ) + l 0 − ( a ℓ ( µ ) − l 0 − 2(  ( µ ) − l 0 ) + 1) , by (48) = l +  ( µ ) − l 0 − 1 < 2 n, b y (42) if a ℓ ( µ ) − l 0 / ∈ 2 Z It remains to pro ve that T is a semistandard tableau. The en tries of S ′ 1 ⊆ S 1 are included in ( T 0 T 1 · · · T l ). More precisely , for i = 1 , . . . ,  ( µ ) − l 0 , a i b elongs to T i , and since S 1 is a symplectic column its row co ordinate is larger or equal than r i . Condition (43) and (44) force the push down of the a i ro w-co ordinates. Hence the column insertion of ( T 0 T 1 · · · T l )) in S ′ ≥ 2 is just the concatenation ( T 0 T 1 · · · T ℓ ( µ ) − l 0 ) .S ′ ≥ 2 . □ 20 OLGA AZENHAS References [ACM09] O. Azenhas, A. Conflitti, and R. Mamede. Linear time equiv alent Littlewoo d–Ric hardson co efficien t maps. Discrete Math. The or. Comput. Sci. Pro c e edings, 21st International Confer ence on F ormal Power Series and A lgebr aic Combinatorics (FPSAC 2009) , pages 127–144, 2009. [ACM25] O. Azenhas, A. Conflitti, and R. Mamede. A uniform action of the dihedral group on Littlewoo d–Ric hardson co effi- cients. , pages 1–59, 2025. [Aze99] O. Azenhas. The admissible interv al for the inv ariant factors of a pro duct of matrices. Line ar Multiline ar Algebr a , (46):51–99, 1999. [Aze25] O. Azenhas. Skew RSK and the switching on ballot tableau pairs. , pages 1–47, 2018, 2025. [Aze26] O. Azenhas. The symplectic left companion of a Littlew o od-Richardson-Sundaram tableau and the Kwon property . arXiv:2601.06930v2 , page 20, 2026. [BZ88] A.D. Berenstein and A.V. Zelevinsky . T ensor pro duct multiplicities and conv ex p olytopes in partition space. J. Ge om. Phys. , 5:453–472, 1988. [F ul97] W. F ulton. Young Table aux with Applic ations to Repr esentation The ory and Ge ometry , volume 35 of Cambridge Univ. Press . London Math. So c. Student T exts, 1997. [GT50] I. M. Gelfand and M.L. Tsetlin. Finite-dimensional representations of the group of unimodular matrices. Dokl. Akad. Nauk SSSR (in R ussian), English tr ansl. in: I.M. Gelfand, Col le cted p ap ers. V ol II, Berlin: Springer-V erlag , (71):653–656, 1950. [GZ85] M. Gelfand and A. V. Zelevinsky . Polyhedra in a space of diagrams and the canonical basis in irreducible represen- tations of g l 3 . F unct. Anal. Appl. , 19:41–144, 1985. [GZ86] I. M. Gelfand and A. V. Zelevinsky . Multiplicities and prop er bases for g l n . Gr oup The or etic al Metho ds in Physics, Pr o c. 3r d Y urmala Seminar 1985, VNU Sci. Press, Utr echt , I I:147–159, 1986. [Kin76] R. C. King. W eight multiplicities for the classical groups. Gr oup the or etic al methods in physics, (F ourth Internat. Col lo q., Nijme gen, 1975), Le ctur e Notes in Phys., Springer, Berlin , V ol. 50:490–499, 1976. [KT25] V. S. Kumar and J. T orres. The branching mo dels of Kwon and Sudaram via flagged hives. Journal of Algebr aic Combinatorics , 62(5):1–14, 2025. [Kwo09] Jae-Hoon Kwon. Cystal graphs and the combinatorics of Young tableaux in M. Hazewinkel (ed.). Handb ook of Algebr a , V ol 6:473–504, 2009. [Kwo18] J.-H. Kwon. Combinatorial extension of stable branching rules for classical groups. T r ansactions of the Americ an Mathematic al So ciety , 370(9):6125–6152, 2018. [Lit44] D. E. Littlewoo d. On in v ariant theory under restricted groups. Philos. T r ans. Roy. So c. L ondon Ser. A, Math. Phys. Sci. , (239–809):387–417, 1944. [LL20] C. Lecouvey and C. Lenart. Combinatorics of generalized exp onen ts. International Mathematics Rese ar ch Notic es , (16):4942–4992, 2020. [NSW26] S. Naito, Y. Suzuki, and H. W atanab e. A proof of the Naito–Sagaki conjecture via the branching rule for ı quantum groups. Journal of Algebr a , 691:32–87, 2026. [Sta98] R. P . Stanley . Enumer ative Combinatorics , volume 62 of Cambridge Studies in Advanc e d Mathematics . Cambridge Universit y Press, Cambridge, 1998. [Sun86] Sheila Sundaram. On the combinatorics of repr esentations of S p (2 n, C ) . PhD thesis . Massachusetts Institute of T echnology . 1986. [Sun90] S. Sundaram. T ableaux in the representation theory of the classical Lie groups. Invariant theory and tableaux (Minne ap olis, MN, 1988) IMA V ol. Math. Appl., 19, Springer, New Y ork, , pages 191–225, 1990. [Tho78] G. P . Thomas. On Sc hensted’s construction and the multiplication of Schur functions. Adv. Math. , 30:8–32, 1978. [W at25a] H. W atanab e. Berele row-insertion and quantum symmetric pairs. , page 32, 2025. [W at25b] H. W atanab e. Symplectic tableaux and quantum symmetric pairs. J. Comb. Algebr a DOI 10.4171/JCA/113 , page 57, 2025. University of Coimbra, CMUC, Dep ar tment of Ma thema tics, Por tugal Email address : oazenhas@mat.uc.pt