Bender-Knuth involutions for types B and C

We show that the combinatorial definitions of King and Sundaram of the symmetric polynomials of types B and C are indeed symmetric, in the sense that they are invariant by the action of the Weyl groups. Our proof is combinatorial and inspired by Bender and Knuth's classic involutions for type A.


Introduction
Symmetric polynomials are a central object of study in two branches of mathematics.On the one side, for combinatorialists, they are generating functions.On the other side, for representation theorists, they are characters of representations.The interplay between these two points of view is best presented with a diagram (see Figure 1).

Generating functions of tableaux
Weyl's character formula

Weight functions of crystals
Cry sta l the ory Je u d e ta q u in Pat h enu me rati on Figure 1: Different approaches to the topic in the literature.
In type A, corresponding to the representation theory of SL  and its combinatorics, the objects and relations in Figure 1 are well understood.Schur (symmetric) polynomials arise as the generating functions of semistandard Young tableaux and as the (Weyl) characters of irreducible representations of SL  .They are also the generating functions of Gelfand-Tsetlin patterns [Sta99].For other Lie groups such as SO 2+1 (type B) and Sp 2 (type C), we have different candidates for tableaux and patterns whose generating functions are the orthogonal polynomials and symplectic polynomials.In this work, we focus on the tableau defined by King for type C [Kin76] and Sundaram for type B [Sun90], who show that their generating functions verify some recursive algebraic formulas and deduce that they recover the irreducible characters of the corresponding Lie groups [Sun86,Sun90].
Characters can be computed as the determinants of matrices whose entries are elementary symmetric polynomials.For type A this is the Jacobi-Trudi formula [Sta99]; for types B and C, see [FH91,KT87].These determinants enumerate tableaux after an argument of Gessel and Viennot for type A [Sta99]; see [FK97] for the type B and C analogues.
On the other hand, crystal bases allow Kashiwara and Nakashima to propose their own tableaux definitions [KN94].For type A, these recover the above combinatorics; for type C, they are seen to be in bijection with King tableaux in [She99], via an analogue of the jeu de taquin algorithm.We are not aware of analogous bijections for type B in the literature.
Without direct proofs, it is not immediately obvious that generating functions of tableaux are the correct candidates for describing characters of representations.In particular, we ask whether these generating functions are invariant by the action of the Weyl group of corresponding type;  ( −1 )   and  (  )   ≀    (  ).This is certainly true of Weyl characters, but we seek a direct and combinatorial proof.
For type A, a short argument by Bender and Knuth [BK72,Sta99] shows that the number of semistandard Young tableaux of a fixed shape  with weight   coincides with the number of weight (   + 1).  for any simple transposition (   + 1),  = 1, ...,  − 1.This is done by constructing an involution  A  which is known as a (type A) Bender-Knuth involution.Therefore, the Schur polynomial of shape  is a symmetric polynomial, living in C[ 1 , ...,   ]   .As a remark, these involutions do not induce an action of   on the set of tableaux of fixed shape, in general.Type A Bender-Knuth involutions are translated to Gelfand-Tsetlin patterns in [BK95]; we review these constructions later in this text.
We introduce type C Bender-Knuth involutions  C  and show combinatorially and directly for the first time that there is an involutory action of  2 ≀   on the set of King patterns of top row .We conclude that symplectic polynomials lie in C[ ± 1 , ...,  ±  ]  2 ≀  .As a corollary, we get type B Bender-Knuth involutions and the analogous result for orthogonal polynomials.The maps are later translated to tableaux.As expected, our involutions do not define an action of  2 ≀   on the sets of patterns of a fixed shape (see [Gut23]).
These results were first claimed by Sundaram in [Sun86].However, as noted by Hopkins [Hop20], the original proof is incorrect and cannot be fixed in any simple way.A corollary of our result, also noted by Sundaram [Sun86,Sun90], is that the symplectic and orthogonal polynomials define class functions on the set of diagonalizable elements in the algebraic groups of types B and C.
A first candidate for type C Bender-Knuth involutions is given as a composition of type A Bender-Knuth involutions.The resulting patterns may not be symplectic, so we postcompose with a rectification map.A 'locality' argument allows us to reduce our proof to computing that  C 2 is an involution on a generic pattern when  = 3.This reduction is what enables us to conclude the result.
It is worth remarking that there are two ways of approaching this computation on generic patterns when  = 3.One is to argue directly and 'by hand', as we do.Alternatively, one could use a computer to check that  C 2 (as a tropical rational map) is involutory.This appears to be beyond the reach of computer algebra systems at the time of writing.But we were able to check that Trop −1  C 2 (as a rational map) is involutory.It remains to argue that the order of  C 2 and the order of Trop −1  C 2 coincide.This step is in general non-trivial (see e.g.[GR16]).
We recall some preliminary definitions in Section 2. We define tableaux and patterns for types A, B, and C in Section 3, and we define symmetric polynomials as their generating functions.We recall type A Bender-Knuth involutions and introduce the type B and C analogues in Section 4. Proving that these are involutions reduces to a computation, that we leave for Section 5.

Preliminary definitions
Fix a natural number  ≥ 1 throughout this work.We work over C. We will follow [Sta99, Ch. 7] for the standard concepts on symmetric polynomials, and [FH91] for Lie theory.

Symmetric polynomials and partitions
The space of symmetric polynomials in  variables is Λ  = C[ 1 , ...,   ]   , where the symmetric group   acts by permuting the variables.A partition  of length ≤  is an -tuple of weakly decreasing non-negative integers.Let Par  be the set of partitions of length ≤ .Bases of Λ  are indexed by Par  .We represent partitions through their Young diagram, which we draw following the English convention.

Lie groups and Weyl groups
The set Irr(SL  ) of irreducible representations of SL  is indexed by Par  .The relationship between Irr(SL  ) and Λ  is explained through the following result: the (Weyl) characters of irreducible representations of SL  form a basis of Λ  .The character of the irreducible representation indexed by  is the Schur polynomial   ( 1 , ...,   ), as defined combinatorially in Section 3.3.
The sets Irr(Sp 2 ) and Irr(SO 2+1 ) are also indexed by Par  .(The set of irreducible representations of the Lie algebra (2 + 1) is richer, and indexed by the set of partitions and half-partitions.The representations indexed by half-partitions are called spin representations, and will not be modelled by the combinatorics of this document.)The irreducible characters for SO 2+1 and Sp 2 are known as the symmetric polynomials of types B and C.These will be defined purely combinatorially in Section 3.3, and referred to as orthogonal polynomials and symplectic polynomials, respectively.Note that these are not in Λ  .Rather, they lie in the ring C[ ± 1 , ...,  ±  ]  of Laurent polynomials invariant under the Weyl group of corresponding type (as a permutation group of the variables).In type A, these Laurent polynomials are polynomials, and the Weyl group of SL  is  ( −1 )   ; this is consistent with the above.The Weyl groups of type B and C coincide, and are isomorphic to the wreath product  2 ≀   .In other words: if we interpret  2 as the permutation group of the set {1, 1 ′ , 2, 2 ′ , ..., ,  ′ }, then  (  ) and  (  ) are isomorphic to the subgroup of  2 generated by (1 1 ′ ) and the permutations 3 Combinatorics and symmetric functions 3.1 Tableaux for types A, B, and C be the set of its cells.Let X be a totally ordered set.A tableau of shape  in the alphabet X is a function  : [] → X.We say a tableau is semistandard if  (, ) <  ( + 1, ) and  (, ) ≤  (,  + 1) whenever this makes sense.
The (set-wise) co-restriction of a map  :  →  to a subset  ⊂  is defined to be the restriction of  to  −1 ().
(B) A (Sundaram) orthogonal tableau  (on  letters) of shape  is a semistandard tableau of shape  in the alphabet A ∞ such that the co-restriction of  to A defines a symplectic tableau (see below), and at most one cell per row takes the value ∞.
(C) A (King) symplectic tableau  (on  letters) of shape  is a semistandard tableau of shape  in the alphabet A such that  (, ) ≥  for all (, ) ∈ [].
We let SSYT  (), SOT  (), and KSpT  () denote the sets of such tableaux.The weight of a tableau  : Example 3.2.We present a semistandard Young tableau, an orthogonal tableau, and a symplectic tableau of shape (3, 3, 2) and their weights.We have a bijection SSYT  () → GT  (), by letting  () be the shape of  −1 [] (see [Sta99]).In other words,  , counts the number of entries ≤  in the th row of  .
Trough this bijection, the th row sum   :=     of a pattern  counts the number of entries ≤  in the corresponding tableau.Therefore, if the weight of a pattern  is defined as the monomial
We think of these as "half-triangular" arrays.We introduce the following definition.
Definition 3.6.A (Sundaram) orthogonal pattern is a symplectic pattern in which top row entries might be circled.Let  be an orthogonal pattern with  rows.The shape  of  is defined by   :=   + 1 if   is circled and   :=   otherwise.For a partition , we let SOP  () be the set of orthogonal patterns with 2 rows and shape .
Trough these bijections, given a pattern  of type B or C, the difference  2−1 − 2−2 counts the number of entries equal to  in the corresponding tableau, whereas  2 −  2−1 counts the number of entries equal to  ′ .Therefore, if the weight of a pattern  is defined as the monomial

Bender-Knuth involutions
We study type B and C analogues of the following elegant proof of Bender and Knuth.

Sketch of proof.
We follow [BK72,Sta99].Let (   + 1) be a simple transposition.Given a tableau of shape  and weight   , we produce a tableau of shape  and weight (   + 1).  by (i) freezing each { ,  + 1}-vertical domino, and (ii) for each row, changing the remaining   (  + 1)  mutable word for   (  + 1)  .This is an involution.■ We denote the th type A Bender-Knuth involution by  A  .It translates to the following map of GT patterns [BK95]: it only affects the th row, and sends where min and max ignore non-existing entries.
We now show the analogue result for type C. We begin by proposing a candidate for type C Bender-Knuth involutions.An involutory action of (1 1 ′ ) on KSpP  () ⊆ GT 2 () is given by  A 1 .For any other generator of  (  ), write this permutation as a product of simple transpositions with respect to the ordered set For each of these, perform a type A involution.
Thanks to the properties of type A Bender-Knuth involutions, the resulting pattern  4 is of weight (   +1)(  ′  +1 ′ ).  0 , as desired.However,  4 needs not be symplectic: we might find an entry   ≠ 0 with 2 > .
Let  = 2, let  >  + 1.By the above, the value of   is min{0, 0} + max{0, 0} − 0 = 0. ■ That is, the only possible obstruction to the symplectic property is the value of entry  +1,2 .We compose with the weight-preserving map rect (rectification) that subtracts  +1,2 from the entries  +1,2 ,  +2,2+1 ,  ,2 , and  ,2−1 , and is the identity everywhere else.Indeed, that is weight-preserving follows from the definition of weight of a pattern, where the variable Define the th type C Bender-Knuth involution as the composite Example 4.6.Let  = 2.We illustrate the 2nd type C Bender-Knuth involution on a symplectic pattern with 6 rows and top row (3, 3, 2).

Proof of Proposition 4.3. It suffices to show that each 𝐵𝐾 C
is an involution.Consider Note Φ is the identity, and that both maps are identical on most entries.Indeed, in each step, the value of  A  on an entry only depends on the value of its four neighbours.Starting with the four entries of  5 that are perturbed by rect, the effect of this perturbation is only measured by the last two entries in rows 2 and 2 ± 1 of  10 .
Therefore, it is enough to show that  C 2 is an involution on a generic pattern with 6 rows.Our strategy to tackle this final computation is to take the entry-wise differences   −  ′  for  = 5, ..., 9. We have  ′ 9 =  0 , and  10 −  ′ 9 is seen to vanish in Section 5. ■ Corollary 4.7.Orthogonal polynomials in  letters are  (  )-symmetric.
To describe type B and C Bender-Knuth involutions on tableaux, it remains to interpret rect.Consider a tableau  0 and the composite Lemma 4.5 says  4 is symplectic up to the existence of { ,  ′ }-vertical dominoes between rows  and  + 1. (For a proof of the lemma in the language of tableaux see [Gut23].)The tableau rect( 4 ) is constructed from  4 by relabelling such dominoes into {  + 1,  + 1 ′ }-vertical dominoes, and sorting rows  and  + 1 into increasing order.
Each of the four first maps are type A Bender-Knuth involutions, and the last map rectifies the tableau by getting rid of the highlighted {2, 2 ′ }-vertical domino.

A computation
To complete the proof of Proposition 4.3, we need to verify that the proposed map  C 2 is an involution on the set of symplectic patterns with 6 rows and fixed shape.
To alleviate notation, we consider a pattern  = ( (6) , ...,  (1) ), and denote with , , , ,  the image of  under the following composite maps: Moreover, we set  as a copy of , and  ′ as a copy of  , and define ,  ′ , ...,  ′ as follows: We have  =  as noted in Section 3.

Figure 2 :
Figure 2: Left: the arrangement of a GT pattern of size 4. Right: the local inequalities.