Descent sets for symplectic groups
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Abstract
The descent set of an oscillating (or updown) tableau is introduced. This descent set plays the same role in the representation theory of the symplectic groups as the descent set of a standard tableau plays in the representation theory of the general linear groups. In particular, we show that the descent set is preserved by Sundaram’s correspondence. This gives a direct combinatorial interpretation of the branching rules for tensor products of the defining representation of the symplectic groups; equivalently, for the Frobenius character of the action of a symmetric group on an isotypic subspace in a tensor power of the defining representation of a symplectic group.
Keywords
Descent set Growth diagram Representations Symmetric functions Quasisymmetric functions Young tableau1 Introduction
In this paper we propose a new combinatorial approach to an open problem in representation theory and solve one particular case. The problem can be formulated in very general terms: let G be a connected reductive complex algebraic group and let V be a finite dimensional rational representation of G. Then, for r≥0, the group G acts diagonally on the tensor power ⊗^{ r } V; moreover the symmetric group, \({\mathfrak{S}}_{r}\), acts by permuting tensor coordinates. Since these two actions commute we have an action of \(G\times{\mathfrak{S}}_{r}\) on ⊗^{ r } V. This representation is completely reducible. The problem is then to describe the decomposition of ⊗^{ r } V into irreducible components. In particular one would like to determine the characters of the isotypic components as representations of \({\mathfrak{S}}_{r}\).
There are two examples we consider: the defining representations of G=GL(n), a general linear group, and G=Sp(2n), a symplectic group. The results for the general linear groups are well known, although not from the point of view taken here, while the results for the symplectic groups are new.
Our approach to this problem builds on the combinatorial theory of crystal graphs. We introduce the notion of descent set of a highest weight vertex in a tensor power of a crystal graph. When V is the defining representation of a general linear group, the usual descent set of a word is a descent set in this sense. We then exhibit a descent set for tensor products of the defining representation of a symplectic group. In particular, our main result, Theorem 4.2 below, is obtained by showing that the Sundaram correspondence for oscillating tableaux preserves descent sets.
The rest of this article is structured as follows. In the next section we will review our setup in the familiar case of the general linear group. Section 3 then considers any complex reductive algebraic group G and contains our new definition of a descent function in this general setting. Section 4 specializes again to the symplectic case and Sect. 5 gives the necessary background about the Sundaram and Berele correspondences. The following section gives the proof of our main result by showing that Sundaram’s bijection preserves descent sets. In the last section, we explore Roby’s description of the Sundaram map using growth diagrams.
2 Robinson–Schensted and the general linear groups
One of the starting points of this project was the remarkable fact, due to Schützenberger [13, Remarque 2], that the usual descent set of a word equals the descent set of its recording tableau under the Robinson–Schensted correspondence. Moreover, the correspondence restricts to a bijection between reverse lattice permutations (also known as reverse Yamanouchi words) of weight μ and the set \(\operatorname{SYT}(\mu)\).
3 The general setting
Let V be a finite dimensional rational representation of a complex reductive algebraic group G. Let Λ be the set of isomorphism classes of irreducible rational representations of G and let V(μ) be the representation corresponding to μ∈Λ. For example, when G is the general linear group GL(n) we can identify Λ with the set of weakly decreasing sequences of integers of length n. Also, when G is the symplectic group Sp(2n) we can identify Λ with the set of partitions with at most n parts. In both cases, the trivial representation corresponds to the empty partition and the defining representation corresponds to the partition 1.
For the symmetric group, \({\mathfrak{S}}_{r}\), we identify the isomorphism classes of its irreducible representations with the set of partitions of r, P(r). We denote the representation corresponding to λ∈P(r) by S(λ).
When V is the defining representation of GL(n) the coefficient a(λ,μ) equals 1 for λ=μ and vanishes otherwise. Thus, in this case the Frobenius character is simply s _{ μ }. For the defining representation of the symplectic group Sp(2n) the coefficients a(λ,μ) were determined by Sundaram [15] and Tokuyama [18]. In general it is a difficult problem to determine these characters explicitly.
We would like to advertise a new approach to this problem, using descent sets. It appears that the proper setting for this approach is the combinatorial theory of crystal graphs. This theory, introduced by Kashiwara, is an offshoot of the representation theory of Drinfeld–Jimbo quantized enveloping algebras. For a textbook treatment we refer to the book by Hong and Kang [5].
For each rational representation V of a connected reductive algebraic group there is a crystal graph. The vector space V is replaced by a set of cardinality dim(V). The raising and lowering operators, which are certain linear operators on V, are replaced by partial functions on the set. It is common practice to represent these partial functions by directed graphs. Each arc is colored according to the application of the Kashiwara operator it represents.
Each vertex of the crystal has a weight and the sum of these weights is the character of the representation, e.g., (3) for the general linear group. Isomorphic representations correspond to crystal graphs that are isomorphic as colored digraphs and the representation is indecomposable if and only if the graph is connected.
A vertex in a crystal graph with no incoming arcs is a highest weight vertex. Each connected component contains a unique highest weight vertex. The weight of this vertex is the weight of the representation it corresponds to. Thus, the components of a crystal graph are in bijection with the set of highest weight vertices.
There is a (relatively) simple way to construct the crystal graph of a tensor product of two representations given their individual crystal graphs. In particular, the highest weight vertices of the crystal corresponding to ⊗^{ r } V can be regarded as words of length r with letters being vertices of the crystal corresponding to V.
For example, when V is the defining representation of GL(n), the highest weight words can be identified with reverse Yamanouchi words, see Definition 5.1. When V is the defining representation of Sp(2n) the highest weight words correspond to nsymplectic oscillating tableaux, see Definition 4.3.
Thus we obtain a combinatorial interpretation of (6) which is a farreaching generalization of the Robinson–Schensted correspondence. Explicit insertion schemes in analogy to the classical correspondence were found corresponding to the defining representation for the symplectic groups Sp(2n) as well as for the odd and even orthogonal groups, see [7].
An important feature of these insertion schemes is that they can be understood as isomorphisms of crystals. In the example of the defining representation of GL(n), the Robinson–Schensted correspondence puts the highest weight words in bijection with standard tableaux. Moreover, the words in each component of the crystal have the same recording tableau, and two words have the same insertion tableau if and only if they occur in the same position of two isomorphic components of the crystal graph.
Let us remark that there are other insertion schemes for the classical groups, for example by Berele [1] for the symplectic groups, Okada [10] for the even orthogonal groups and Sundaram [16] for the odd orthogonal groups. However, these are not isomorphisms of crystals.
We can now state the fundamental property we require for a descent set in the general sense.
Definition 3.1
Thus, by (5), the usual descent set of a word is a descent set in the sense of this definition for the defining representation of GL(n). We note that in terms of the crystal graph (9) above, a highest weight vertex w _{1} w _{2}⋯w _{ r } has a descent at position k if and only if there is a (nontrivial) directed path from w _{ k+1} to w _{ k } in the crystal graph.
In this article we exhibit a descent function with a similar description for the symplectic groups, see Definition 4.1.
4 Oscillating tableaux and descents
In the case of the defining representation of the symplectic group Sp(2n) the vertices of the crystal graph corresponding to ⊗^{ r } V are words w=w _{1} w _{2}⋯w _{ r } in {±1,…,±n}^{ r }. The weight of a vertex is the tuple \(\operatorname{wt}(w)=(\mu_{1},\dots,\mu_{n})\), where μ _{ i } is the number of letters i minus the number of letters −i in w. The vertex is a highest weight vertex if for any k≤r, the weight of w _{1}⋯w _{ k } is a partition, i.e., μ _{1}≥μ _{2}≥⋯≥μ _{ n }≥0.
Definition 4.1
A highest weight vertex w _{1} w _{2}⋯w _{ r } in the crystal graph corresponding to ⊗^{ r } V has a descent at position k if there is a (nontrivial) directed path from w _{ k } to w _{ k+1} in the crystal graph (10).
We can now state our main result.
Theorem 4.2
To prove this theorem we will first rephrase it in terms of nsymplectic oscillating tableaux, also known as updowntableaux, which are in bijection with the highest weight vertices in the crystal graph corresponding to ⊗^{ r } V.
Definition 4.3
The kth step, going from μ ^{ k−1} to μ ^{ k }, is an expansion if a box is added and a contraction if a box is deleted. We will refer to the box that is added or deleted in the kth step as b _{ k }.
The oscillating tableau \(\mathcal{T}=(\mu^{0},\mu^{1},\ldots,\mu^{r})\) is nsymplectic if every partition μ ^{ i } has at most n nonzero parts.
The next result follows immediately from the definitions above.
Proposition 4.4
 given a highest weight vertex w _{1} w _{2}⋯w _{ r }, the oscillating tableau is the sequence of weights of its initial factors$$\bigl(\operatorname{wt}(w_1), \operatorname{wt}(w_1w_2), \operatorname {wt}(w_1w_2w_3),\dots, \operatorname{wt}(w_1 w_2\cdots w_r)\bigr); $$

for an oscillating tableau \(\mathcal{T}\) the corresponding word w _{1} w _{2}⋯w _{ r } is given by w _{ k }=±i, where i is the row of b _{ k } and one uses plus or minus if b _{ k } is added or deleted, respectively.
Note that in general, the oscillating tableau obtained via Berele’s correspondence (see Sect. 5) from a word w _{1} w _{2}⋯w _{ r } is different from the oscillating tableau given by the bijection above.
Example 4.5
We will now define the descent set of an oscillating tableau in such a way that a tableau \(\mathcal{T}\) and its word w will always have the same descent set.
Definition 4.6

step k is an expansion and step k+1 is a contraction, or

steps k and k+1 are both expansions and b _{ k } is in a row strictly above b _{ k+1}, or

steps k and k+1 are both contractions and b _{ k } is in a row strictly below b _{ k+1}.
Example 4.7
The fact that these two descent sets always coincide is easy to prove directly from the definitions, so we omit the proof and just formally state the result.
Proposition 4.8
The descent set of an nsymplectic oscillating tableau of length r coincides with the descent set of the corresponding highest weight vertex of the crystal graph of ⊗^{ r } V.
Thus it suffices to prove the following variant of Theorem 4.2.
Theorem 4.9
In Sundaram’s correspondence, an arbitrary oscillating tableau \(\mathcal{T}\) is first transformed into a fixedpointfree involution ι on a subset A of the positive integers and a partial Young tableau T, that is, a filling of a Ferrers shape with all entries distinct and increasing in rows and columns. There are natural notions of descents for these objects which extend those for permutations in \({\mathfrak{S}}_{r}\) and standard Young tableaux.
Definition 4.10
The definition of the descent set for an oscillating tableau is constructed so that it contains the union of the descent sets of the associated partial tableau and involution under Sundaram’s bijection.
5 The correspondences of Berele and Sundaram
One of our main tools for proving Theorem 4.9 will be a bijection, \(\operatorname{Sun}\), due to Sundaram [15, 17]. Berele [1] constructed a bijection which is a combinatorial counterpart of the isomorphism in (6) when V is the defining representation of Sp(2n) In combination with Berele’s correspondence, Sundaram’s bijection can be regarded in its turn as a combinatorial counterpart of the isomorphism in (7). In this section we define the objects involved; the bijection \(\operatorname{Sun}\) itself will be described in detail in the next section.
Definition 5.1
Let u be a word with letters in the positive integers. Then u is a Yamanouchi word (or lattice permutation) if, in each initial factor u _{1} u _{2}⋯u _{ k }, there are at least as many occurrences of i as there are of i+1 for all i≥1. The weight β of a lattice permutation u is the partition β=(β _{1}≥β _{2}≥⋯), where β _{ i } is the number of occurrences of the letter i in u.
For a skew semistandard Young tableau S the reading word, w(S), is the word obtained by concatenating the rows from bottom to top. It has the nice property that applying the Robinson–Schensted map to w(S) one recovers S as the insertion tableau. We will need the reverse reading word, \(w^{\operatorname{rev}}(S)\), which is obtained by reading w(S) backwards.

its reverse reading word is a lattice permutation of weight β, where β is a partition with all columns having even length, and

entries in row n+i+1 of S are greater than or equal to 2i+2 for i≥0.
We will denote the number of nsymplectic Littlewood–Richardson tableaux of shape λ/μ and weight β by \(c_{\mu,\beta}^{\lambda}(n)\).
For ℓ(λ)≤n+1 we see that the number \(c_{\mu,\beta }^{\lambda}(n)\) is the usual Littlewood–Richardson coefficient whenever β is a partition with all columns having even length. This is trivial for ℓ(λ)≤n, and follows from the correspondence \(\operatorname{Sun}\) described below for ℓ(λ)=n+1.
We now have all the definitions in place to explain the domain and range of the correspondence \(\operatorname{Sun}\).
Theorem 5.2
([15, Theorem 9.4])

Q is a standard tableau of shape λ, with λ=r, and

S is an nsymplectic Littlewood–Richardson tableau of shape λ/μ and weight β, where β has even columns and β=r−μ.
For completeness, let us point out the relation between Sundaram’s bijection and Berele’s correspondence. This correspondence involves the following objects due to King [6], indexing the irreducible representations of the symplectic group Sp(2n).
Definition 5.3

entries in rows are weakly increasing,

entries in columns are strictly increasing, and

the entries in row i are greater than or equal to i, in the above ordering.
Now consider an nsymplectic oscillating tableau as a word in the ordered alphabet 1<−1<2<−2<⋯<n<−n, as described just after Definition 4.3. We can then apply the Robinson–Schensted correspondence to obtain a semistandard Young tableau \(P_{\operatorname{RS}}\) in this alphabet and an (ordinary) standard Young tableau \(Q_{\operatorname{RS}}\). Alternatively, we can compose Berele’s correspondence with Sundaram’s bijection to obtain a triple \((P_{\operatorname{Ber}},Q_{\operatorname{Sun}},S_{\operatorname {Sun}})\). It then turns out that \(Q_{\operatorname{RS}}=Q_{\operatorname{Sun}}\). This implies that for each standard Young tableau \(Q_{\operatorname{Sun}}\) we have a correspondence \(P_{\operatorname{RS}}\mapsto (P_{\operatorname{Ber}}, S_{\operatorname{Sun}})\). Moreover, this correspondence is independent of the choice of \(Q_{\operatorname{Sun}}\). One can then prove the following theorem.
Theorem 5.4
([15, Theorem 12.1])
Let us point out a corollary, which settles a conjecture from [19].
Corollary 5.5
Proof
Comparing (8) with the previous Theorem, it suffices to show that \(c_{\emptyset,\beta}^{\lambda}\) is nonzero only when λ=β is a partition of the type described in the corollary, and in this case there is only one corresponding tableau. Suppose S is an nsymplectic Littlewood–Richardson tableau of shape λ/∅ and weight β. Then S is a semistandard Young tableau of straight shape λ whose reverse reading word is a lattice permutation. This implies that all entries in row j of S are equal to j. In particular, λ=β and β satisfies the conditions of Theorem 5.4. Furthermore, it is required that the entries in row j=n+i+1 are at least 2i+2 for i≥0. It follows that i+1≤n, and therefore j≤2n. □
6 Proof of the main result
In light of Sundaram’s results, we claim that to prove Theorem 4.9 it suffices to demonstrate the following.
Theorem 6.1
Proof of Theorem 4.9
6.1 Sundaram’s first bijection
We now describe Sundaram’s first bijection which we will denote by \(\operatorname{Sun}_{1}\). It maps an oscillating tableau \(\mathcal{T}\) to a pair (ι,T) where ι is a fixedpointfree involution, T is a partial Young tableau, and the entries of ι and T are complementary sets.
Let \(\mathcal{T}=(\emptyset=\mu^{0},\mu^{1},\dots,\mu^{r})\) be an oscillating tableau. We then construct a sequence of pairs (ι _{ k },T _{ k }) for 0≤k≤r such that T _{ k } has shape \(\operatorname{sh}(T_{k})=\mu^{k}\), and the entries of ι _{ k } and of T _{ k } form a set partition of {1,…,k}.

If the kth step is an expansion then ι _{ k }=ι _{ k−1} and T _{ k } is obtained from T _{ k−1} by putting k in box b _{ k }.

If the kth step is a contraction then take T _{ k−1} and bump out (using Robinson–Schensted column deletion) the entry in box b _{ k } to get a letter x and the partial Young tableau T _{ k }. The involution ι _{ k } is ι _{ k−1} with the transposition (x,k) adjoined.
Lemma 6.2
(Sundaram [15, Lemma 8.7])

ι is a fixedpointfree involution of a set A⊆{1,…,r}, and

T is a partial tableau of shape μ such that its set of entries is {1,…,r}∖A.
In general it seems that, given the pair (ι,T), there is no straightforward way to determine whether the corresponding oscillating tableau is nsymplectic, see [15, Lemma 9.3]. However, when μ=∅ the map \(\operatorname{Sun}_{1}\) is in fact a bijection between nsymplectic oscillating tableaux and (n+1)nonnesting perfect matchings of {1,2,…,r}, see [2].
Example 6.3
We can now take our first step in proving Theorem 6.1.
Proposition 6.4
Proof
We proceed by analysing the effect of two successive steps in the oscillating tableau.
If step k is an expansion and step k+1 is a contraction then k+1∈ι and ι(k+1)<k+1. Now k either ends up in T or in ι. In the former case, \(k\in\operatorname{Des}(T/\iota )\). In the latter case ι(k)>k≥ι(k+1) and \(k\in\operatorname {Des}(\iota)\). In both cases this gives a descent of (ι,T).
If step k is a contraction and step k+1 is an expansion then k∈ι and ι(k)<k. Now either k+1 ends up in T or in ι. In the former case \(k\in\operatorname{Des}(\iota/T)\) rather than \(\operatorname{Des}(T/\iota)\). In the latter case ι(k)<k<k+1<ι(k+1). Neither of these cases gives a descent of (ι,T).
If steps k and k+1 are both contractions then k,k+1∈ι. If b _{ k } is strictly below b _{ k+1} then, by wellknown properties of \(\operatorname{RS}\), the element removed when bumping out b _{ k } will be in a lower row than the one obtained when bumping out b _{ k+1}. Thus ι(k)>ι(k+1) and \(k\in\operatorname{Des}(\iota,T)\) as desired. By a similar argument, if b _{ k } is weakly above b _{ k+1} then ι(k)<ι(k+1) and \(k\not\in\operatorname{Des}(\iota,T)\).
Now suppose steps k and k+1 are both expansions. If b _{ k } is in a row strictly above b _{ k+1}, then any column deletion will keep k in a row strictly above k+1. It follows that at the end we have one of three possibilities. The first is that k,k+1∈T and, as was just observed, we must have \(k\in\operatorname{Des}(T)\). If either element is removed, then k+1 must be removed first because the row condition forces k+1 to always be in a column weakly left of k. So at the end we either have k∈T and k+1∈ι, or we have ι(k)>ι(k+1). So in every case \(k\in\operatorname {Des}(\iota,T)\). By a similar argument, if b _{ k } is in a row weakly below b _{ k+1}, then \(k\not\in\operatorname{Des}(\iota,T)\).
This completes the check of all the cases and the proof. □
6.2 Robinson–Schensted
The next step of the map \(\operatorname{Sun}\) is to apply the Robinson–Schensted correspondence to the fixedpointfree involution ι to obtain a partial Young tableau I with the same set of entries as ι. Let us first recall the following wellknown facts about the Robinson–Schensted map.
Lemma 6.5
([12, Exercise 22a] and [14, Exercise 7.28a])

\(\operatorname{Des}(\pi)=\operatorname{Des}(Q)\), and

π is a fixedpointfree involution if and only if P=Q and all columns of Q have even length.
Combining Proposition 6.4 with this lemma we obtain the following result.
Proposition 6.6
Example 6.7
6.3 Sundaram’s second bijection
To finish defining \(\operatorname{Sun}\), we need a bijection \(\operatorname{Sun}_{2}\) that transforms the pair of partial Young tableaux (I,T) to a pair (Q,S) as in Theorem 5.2.
Let Q be the standard Young tableau of shape λ obtained by column inserting the reverse reading word of the tableau I into the tableau T. Also construct a skew semistandard Young tableau S as follows: whenever a letter of \(w^{\operatorname{rev}}(I)\) is inserted, record its row index in I in the box which is added. Then \(\operatorname{Sun}_{2}(I, T) = (Q, S)\). Sundaram actually defines this map for pairs (I,T) of semistandard Young tableaux. But we will not need this level of generality.
Lemma 6.8
(Sundaram [15, Theorem 8.11, Theorem 9.4])

Q is a standard Young tableau of shape λ∈P(r) and

S is a semistandard tableau of shape λ/μ and weight β.
Example 6.9
To prove Theorem 6.1 we need two properties of column insertion.
Lemma 6.10
Proof of Theorem 6.1
By the second assertion of Lemma 6.10 we find that Q can also be obtained by column inserting the concatenation of \(w^{\operatorname{rev}}(T)\) and \(w^{\operatorname{rev}}(I)\). By the first assertion of Lemma 6.10, the descent set of Q equals Open image in new window , which in turn equals the descent set of \(\mathcal{T}\) by Proposition 6.6. □
7 Growth diagrams and Roby’s description of \(\operatorname{Sun}\)
Using Fomin’s framework of growth diagrams [3, 4], Roby [11] showed how to obtain the standard tableau Q directly from the oscillating tableau \(\mathcal{T}\). However, he omitted the proof that his construction does indeed correspond with that of Sundaram’s. We take the opportunity here to review Roby’s construction and provide this proof. Using his example, Roby also attempted to obtain a direct description of the nsymplectic Littlewood–Richardson tableau. However, he made the assumption that a permutation can be partitioned into disjoint increasing subsequences whose lengths are given by the lengths of the rows of the standard Young tableaux associated via the Robinson–Schensted correspondence, which is not true in general.
In the following we use the notation λ⋖μ to denote that λ is covered by μ. Furthermore, for an integer i and a partition λ we denote by λ+ϵ _{ i } the partition obtained from λ by adding one to the ith part, assuming that λ _{ i }<λ _{ i−1}. Similarly we define λ−ϵ _{ i }.
 F1
If μ≠ν, then let ρ=μ∪ν.
 F2
If λ⋖μ=ν then we must have μ=λ+ϵ _{ i } for some i, so let ρ=μ+ϵ _{ i+1}.
 F3If λ=μ=ν, then let$$\rho= \begin{cases} \lambda&\text{if $c$ does not contain an $X$,}\\ \lambda+\epsilon_1 &\text{if $c$ contains an $X$.} \end{cases} $$
 B1
If μ≠ν, then let λ=μ∩ν.
 B2If ρ⋗μ=ν then we must have μ=ρ−ϵ _{ i } for some i, so letwhere c gets filled with an X in the first case and is left empty in the second.$$\lambda= \begin{cases} \mu&\text{if $i=1$,}\\ \mu\epsilon_{i1} &\text{if $i\ge2$,} \end{cases} $$
 B3
If ρ=μ=ν then let λ=μ.
The local rules are, in fact, just another way to perform Robinson–Schensted insertion. To explain this, consider sequences of partitions ϵ=(ϵ _{0},ϵ _{1},…,ϵ _{ r }) where ϵ _{0}=∅ and, for k>0, ϵ _{ k } either equals ϵ _{ k−1} or is obtained from ϵ _{ k−1} by adding a box. Such a sequence ϵ corresponds to a partial Young tableau P in the same way that an oscillating tableau with all steps expansions corresponds to a standard Young tableau: if ϵ _{ k }=ϵ _{ k−1} then k does not appear in P; and if ϵ _{ k } is obtained by adding a box to ϵ _{ k−1} then k appears in P in the box added. Note that if the border of any row in a growth diagram starts with the empty partition, then the sequence of diagrams along this border satisfies the conditions for such an ϵ.
Theorem 7.1
(see [14, Theorem 7.13.5])
Consider a growth diagram consisting of a single row whose cell in column x contains a cross and whose left border is labeled with empty partitions. If the lower and upper borders are identified with partial tableaux P and P′, respectively, then P′ is obtained by row inserting x into P.
Note that, usually, this theorem is stated with the roles of columns and rows interchanged. However, this is without consequence since the local rules are symmetric in this respect.
 R1
Label the corners of the cells along the main diagonal, i.e., from northwest to southeast, of an r×r grid with the conjugates of these partitions.
 R2
Using the rules B1–B4, determine the partitions labeling the corners of the cells below the main diagonal and which of these cells contain a cross. (Since neighboring partitions on the diagonal will always be distinct, rule B1 will always apply to determine the subdiagonal without needing to know ρ.)
 R3
Place crosses into those cells above the main diagonal, whose image under reflection about the main diagonal contains a cross.
 R4
Using the rules F1–F4, compute the partitions labeling the corners of the cells above the main diagonal.

\(A_{\operatorname{Rob}}\) be the set of column indices of the growth diagram that contain a cross where indices are taken as in a matrix,

\(\iota_{\operatorname{Rob}}\) be the involution of \(A_{\operatorname{Rob}}\) that exchanges column and row indices of the cells containing crosses,

\(T_{\operatorname{Rob}}\) be the partial Young tableau defined by \((\emptyset=\tau_{0}^{t},\dots,\tau_{r}^{t})\), and

\(Q_{\operatorname{Rob}}\) be the partial Young tableau defined by \((\emptyset=\kappa_{0}^{t},\dots,\kappa_{r}^{t})\).
We now compute a second growth diagram. Consider a square grid having crosses in the same places as determined before. Then, label the corners along the lower and the left border of the grid with the empty partition. Finally, using the rules F1–F4 compute the sequence of partitions (∅=ν _{0},ν _{1},…,ν _{ r }), labeling the corners along the upper border of the growth diagram. Let \(I_{\operatorname{Rob}}\) be the partial Young tableau defined by \((\emptyset=\nu_{0}^{t},\dots,\nu_{r}^{t})\).
Comparing these objects with the ones computed using Sundaram’s correspondence, the reader will have anticipated the following theorem.
Theorem 7.2
As mentioned at the beginning of this section it appears that there is no straightforward way to extract the skew Littlewood–Richardson tableau S from the growth diagram.
For the proof the following simple observation will be helpful.
Lemma 7.3
Suppose that the left border of a growth diagram is labeled by empty partitions and let c be a cell labeled with partitions as in Fig. 1. Then λ=μ if and only if none of the cells to the left of c and in the same row as c contains a cross.
Proof of Theorem 7.2
Let us first show that \(A_{\operatorname{Rob}}=A\), \(\iota _{\operatorname{Rob}}=\iota\) and \(T_{\operatorname{Rob}}=T\). We will use induction on the length r of the oscillating tableau \(\mathcal{T}\). There is nothing to show if \(\mathcal{T}\) is empty. Thus, suppose that r>0 and that \(\mathcal{T}\) with the last step removed is mapped to the pair (ι′,T′) where ι′ is an involution on A′.
If the last step of \(\mathcal{T}=(\mu_{0},\ldots,\mu_{r})\) is an expansion, then we claim that the bottom row of the growth diagram does not contain a cross. Since μ _{ r−1}⋖μ _{ r } the partition τ _{ r−1} is obtained by applying rule B1 and we have τ _{ r−1}=μ _{ r−1}. Now the claim follows from the lemma.
We now have \(A_{\operatorname{Rob}}=A'=A\) and \(\iota_{\operatorname {Rob}} = \iota' = \iota\) in agreement with Sundaram’s first correspondence \(\operatorname{Sun}_{1}\). Moreover, since we have τ _{ r−1}=μ _{ r−1}⋖μ _{ r }=τ _{ r }, \(T_{\operatorname{Rob}}\) is obtained from T′ by putting r into the cell added by the expansion, which coincides with T as constructed via \(\operatorname{Sun}_{1}\).
If the last step of \(\mathcal{T}\) is a contraction, again using the lemma, the rule B1 entails that there must be a cross in the bottom row of the growth diagram, say in column x. Thus, \(\iota_{\operatorname {Rob}}\) is obtained from ι′ by adjoining the pair (x,r). It remains to show that \(T_{\operatorname{Rob}}\) is obtained by a column deletion of T′ where x is the element removed at the end of the deletion process. Thus we must show that column inserting x into T yields T′, or equivalently, that row inserting x into T ^{ t } yields (T′)^{ t }. This last statement follows immediately from Theorem 7.1.
We now turn to the computation of Q and \(Q_{\operatorname{Rob}}\). Using \(\operatorname{Sun}_{2}\) the tableau Q is obtained by column inserting the reverse reading word of I into T. Equivalently, Q ^{ t } is obtained by row inserting \(w^{\operatorname{rev}}(I)\) into T ^{ t }. Because \(w^{\operatorname{rev}}(I)\) is Knuth equivalent to the reversal of the involution ι in one line notation, Theorem 7.1 shows that the tableau described by the partitions along the upper border of the growth diagram is indeed Q ^{ t }.
The equality of I and \(I_{\operatorname{Rob}}\) also follows directly from Theorem 7.1. □
One nice feature of the growth diagram formulation of Sundaram’s correspondence is that the descent set can be visualized. Namely, take any partial permutation with insertion tableau T ^{ t } and construct its growth diagram. On top of this stack the growth diagram for the oscillating tableau so that the two rows corresponding to T ^{ t } coincide. Then k is a descent in the oscillating tableau if and only if the cross in column k is lower than the cross in column k+1. It is in fact possible to prove Theorem 6.1 using this approach.
Notes
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