On Classical groups detected by the triple tensor product and the Littlewood–Richardson semigroup
 Heekyoung Hahn^{1}Email author
DOI: 10.1007/s4099301600493
© The Author(s) 2016
Received: 18 February 2016
Accepted: 5 July 2016
Published: 3 October 2016
Abstract
Langlands’ beyond endoscopy proposal for establishing functoriality motivates the study of irreducible subgroups of \(\mathrm {GL}_n\) that stabilize a line in a given repesentation of \(\mathrm {GL}_n\). Such subgroups are said to be detected by the representation. In this paper we continue our study of the important special case where the representation of \(\mathrm {GL}_n\) is the triple tensor product representation \(\otimes ^3\). We prove a family of results describing when subgroups isomorphic to classical groups of type \(B_{n}\), \(C_n\), \(D_{2n}\) are detected.
Mathematics Subject Classification
Primary 11F70 Secondary 20G051 Background
One can ask for a characterization of those automorphic representations in the image. To explain Langlands’ conjectural criterion for an automorphic representation to be in the image, we recall the following definition from [11].
Definition 1.1
Let H be an irreducible reductive subgroup of \(\mathrm {GL}_N\). We say a representation r : \(\mathrm {GL}_N \longrightarrow \mathrm {GL}(V)\) detects H if H stabilizes a line in V.
Remark
If H is connected then r detects H if and only if it detects \(H^{\mathrm {der}}\).
Let \({}^{\lambda }H\) denote the Zariski closure in \(\mathrm {GL}_N(\mathbb {C})\) of the image of \({}^LH\) under the map (1.1). The following conjecture is the crux of Langlands’ beyond endoscopy proposal [21], which aims to prove Langlands functoriality in general:
Conjecture 1.2
Let \(\pi \) be a unitary cuspidal automorphic representation of \(\mathrm {GL}_N(\mathbb {A}_F).\) If \(\pi \) is a functorial transfer from H , then \(L(s,\pi ,r \otimes \chi )\) has a pole at \(s=1\) for some character \(\chi \in F^{\times } \backslash \mathbb {A}_F^{\times } \rightarrow \mathbb {C}^\times \) whenever r detects \({}^{\lambda }H\).
Motivated by Langlands’ proposal, in the recent paper [11], the author proposed the following concrete question in algebraic group theory:
Question 1.3
If \(r=\mathrm {Sym}^2\), one knows that every irreducible reductive subgroup of \(\mathrm {GL}_N\) detected by r is conjugate to a subgroup of \(\mathrm {GO}_N\). Moreover, in this case Conjecture 1.2 is proven by work of Arthur [2], work of Cogdell et al. [5] and work of Ginzburg et al. [10]. There is a similar statement for \(r=\Lambda ^2\). Thus the case \(r=\otimes ^2\) is relatively wellunderstood.
Apart from this special case, explicit results are hard to come by (but see [9]). In [11] we initiated the study of the subgroups of \(\mathrm {GL}_N\) detected by \(\otimes ^3\) by studying irreducible simple subgroups of type \(A_n\). In this paper we continue this investigation for the series of classical groups.

(A1) The algebraic group G is one of the classical groups \(\mathrm {SO}(2n+1)\), \(\mathrm {Sp}(2n)\) and \(\mathrm {SO}(2n)\), where for \(\mathrm {SO}(2n)\), we further assume n is even.

(A2) For \(G=\mathrm {SO}(2n)\), we assume that \(\lambda _n=0\).
We now state our results on detection. We start by observing that the property of \(\mathbb {S}_{\parallel \lambda \parallel }(G)\) being detected is essentially stable under replacing \(\lambda \) by its conjugate partition \(\lambda '\) (see Sect. 2.1 for the definition):
Theorem 1.4
Let \(\lambda \in P_n\) be a partition such that its conjugate \(\lambda '\) is also in \(P_n\). Then \(\mathbb {S}_{\parallel \lambda \parallel }(G)\) is detected by \(\otimes ^3\) if and only if \(\mathbb {S}_{\parallel \lambda '\parallel }(G)\) is detected by \(\otimes ^3\).
If \(\lambda \) is a partition of an odd number, then \(\mathbb {S}_\lambda (G)\) is never detected:
Theorem 1.5
Let \(\lambda \in P_n\) be a partition of an odd number. Then the representation \(\otimes ^3\) does not detect \(\mathbb {S}_{\parallel \lambda \parallel }(G)\).
Conversely, we prove the following result which constructs families of subgroups that are detected by \(\otimes ^3\).
Theorem 1.6
 (a)
all \(\lambda _i\) are even, or
 (b)
it has even number of nonzero \(\lambda _i\) and all nonzero \(\lambda _i\) are distinct and odd, or
 (c)
\(\lambda \) is a hook partition,
To experts in algebraic combinatorics and representation theory it is clear that the results above ought to have something to do with the famous Littlewood–Richardson semigroup \(\mathrm {LR}_n\) of order n (see Sect. 3 for definition) or equivalently with the Littlewood–Richardson coefficients \(c^\lambda _{\mu \nu }\) (see Sect. 2.2 for the definition). Studying \(c^{\lambda }_{\mu \nu }\) or \(\mathrm {LR}_n\) is a central topic in the representation theory (see [7, 22, 23] and [27], for example), in combinatorics of symmetric functions ([23, 24, 26]), in the topology Grassmann varieties and hence the theory of vector bundles and Ktheory [14]. Also there has been a long history in connection to Hermitian eigenvalues and Horn’s conjecture ([12] and see [16] for example). It is wellknown that \(\mathrm {LR}_n\) has saturation property, namely, if \((k\lambda , k\mu , k\nu )\in \mathrm {LR}_n\) for some \(k>0\), then \((\lambda , \mu , \nu )\in \mathrm {LR}_n\). This is a famous theorem of Knudson and Tao [17] (see also [18]). For various descriptions of \(\mathrm {LR}_n\) we refer to [4, 13, 28]. The precise connection between \(\mathrm {LR}_n\) and detection via \(\otimes ^3\) is given in the following theorem, which is essentially a reformulation of the Newell–Littlewood formula (2.6):
Theorem 1.7
The subgroup \(\mathbb {S}_{\parallel \lambda \parallel }(G)\) is detected by \(\otimes ^3\) if and only if there are \(\alpha , \beta , \gamma \in P_n\) such that all triples \((\lambda , \alpha , \beta )\), \((\lambda , \alpha , \gamma )\) and \((\lambda , \beta , \gamma )\) are elements in \(\mathrm {LR}_n\).
Theorems 1.4 and 1.5 follow formally from Theorem 1.7. However, in general, even given Theorem 1.7, it is not clear how to use the descriptions of the Littlewood–Richardson coefficients given in the references above to describe which subgroups \(\mathbb {S}_{\parallel \lambda \parallel }(G)\) are detected by \(\otimes ^3\). In particular our proof of Theorem 1.6 requires some “hands on” combinatorics with partitions.
Remark
Given the saturation theorem for the Littlewood–Richardson semigroup \(\mathrm {LR}_n\), it is natural to ask whether \(\mathbb {S}_{\parallel k\lambda \parallel }(G)\) is detected by \(\otimes ^3\) for some \(k>0\) implies that \(\mathbb {S}_{\parallel \lambda \parallel }(G)\) is detected. This is evidently false. Indeed, Theorem 1.5 implies that if \(\lambda \) is a partition of an odd number then \(\mathbb {S}_{\parallel \lambda \parallel }(G)\) is not detected. However, \(\mathbb {S}_{\parallel k\lambda \parallel }(G)\) is always detected if k is even by Theorem 1.6 (a).
Before outlining the paper we comment on these results from the perspective of Langlands’ beyond endoscopy proposal. Essentially they give some feeling of how much more complicated the structure of the beyond endoscopy proposal is than the theory of endoscopy. In the theory of endoscopy one writes the trace formula in terms of stable orbital integrals on endoscopic groups. In the beyond endoscopy proposal Langlands proposes that one writes limiting forms of the trace formula in terms of groups that are detected by a particular representation. It is evident from the theorems above that this set is much more complicated than the set of endoscopic groups of a given group. On the other hand there may be simplifications that can be made in certain situations. For example, all of the groups \(\mathbb {S}_{\parallel \lambda \parallel }(G) \leqslant \mathrm {GL}_N\) considered above are conjugate to subgroups of \(\mathrm {O}_N\) or \(\mathrm {Sp}_N\) since the representations \(\mathbb {S}_{\parallel \lambda \parallel }\) are selfdual. Thus for some purposes it might be possible to sieve them all out at the outset by restricting to nonself dual representations.
We close our introduction by outlining the paper. In Sect. 2, we review some basic facts on partitions, the Littlewood–Richardson coefficients and the groups \(\mathrm {SO}(2n+1)\), \(\mathrm {Sp}(2n)\) and \(\mathrm {SO}(2n)\). In Sect. 3, we prove a key proposition showing a necessary and sufficient condition for detection by \(\otimes ^3\) and discuss the connection to the Littlewood–Richardon semigroup \(\mathrm {LR}_n\) and prove Theorems 1.4, 1.5 and 1.7. In Sect. 4, we prove Theorem 1.6 by constructing \(\lambda \) explicitly such that \(\mathbb {S}_{\parallel \lambda \parallel }(G)\) is detected by \(\otimes ^3\).
2 Preliminaries
2.1 Partitions
In this section, we recall some basic notion in the theory of partitions. For \(\lambda \in P_n\), we let \(\lambda =\sum _i\lambda _i\) be the number partitioned by \(\lambda \). Moreover, denote by \(\ell (\lambda )\) the number of nonzero \(\lambda _i\) for a partition \(\lambda \).
The conjugate of a partition \(\lambda \) is the partition of \(\lambda \) whose Young diagram is obtained by reflecting the Young diagram of \(\lambda \) about the diagonal so that rows become columns and columns become rows. We write it as \(\lambda '\). In above example \(\lambda =(5,3,1)\), the conjugate partition of \(\lambda \) is \(\lambda '=(3,2,2, 1,1)\) (see [1] for instance).
2.2 Littlewood–Richardson coefficients
In this section, we follow the exposition of [8] as we recall basic facts: For any n dimensional vector space V over \(\mathbb {C}\) and any partition \(\lambda \in P_n\), we can apply the Schur functor \(\mathbb {S}_{\lambda }\) to V to obtain a representation \(\mathbb {S}_{\lambda }(V)\) for \(\mathrm {GL}_n\). It remains irreducible when restrict to \(\mathrm {SL}_n\). In particular it determines an irreducible representation of the Lie algebra \(\mathfrak {sl}_n\) (see [8, Proposition 15.15]).
As our proofs for Theorem 1.6 rely heavily on this rule, we will briefly state it following the exposition of [13, §4]. Let \(\mu =(\mu _1, \mu _2, \dots , \mu _n),\,\lambda =(\lambda _1, \lambda _2, \dots , \lambda _n)\in P_n\). One writes \(\mu \subset \lambda \) if the Young diagram of \(\mu \) sits inside the Young diagram of \(\lambda \). In other words, \(\mu _i\leqslant \lambda _i\) for all i. If \(\mu \subset \lambda \), put the Young diagram of \(\mu \) on the Young diagram of \(\lambda \) with the same topleft corner and remove \(\mu \) out of \(\lambda \). That way, we obtain the skew diagram \(\lambda \mu \). Put a positive number in each box \(\lambda \mu \), then it becomes a skew tableau with the shape \(\lambda \mu \). If the entries of this skew tableau are taken from \(\{1, 2, \dots , n\}\) and \(\nu _j\) of them are j for each \(j=1, 2, \dots , m\), then the content of this skew tableau becomes \(\nu =(\nu _1, \dots , \nu _n)\). For a skew tableau T, we define the word of T by the sequence w(T) of positive integers obtained by reading the entries of T from top to bottom and right to left in each row.
Definition 2.1
 (i)
The numbers in each row of T weakly increase from left to right and the numbers in each column of T strictly increase from top to bottom.
 (ii)
For each positive integer j, starting from the first entry of w(T) to any place in w(T), there are at least as many as j’s as \((j+1)\)’s.
2.3 Classical groups \(\mathrm {SO}(2n+1)\), \(\mathrm {Sp}(2n)\) and \(\mathrm {SO}(2n)\)
In this section, we briefly go over the basic facts on orthogonal and symplectic groups which we will be using mainly for the purpose of fixing notations. We follow the expositions of [8] in part.
Let G be the subgroup of \(\mathrm {SL}(V)\) which preserves this form. Then by using the CartanKilling classification, one knows that G is of type \(B_n\) (resp. type \(D_n\)) if Q is orthogonal and \(\dim V=2n+1\) (resp. \(\dim V=2n\)). In the former case we write \(G=\mathrm {SO}(2n+1)\) and in the latter we write \(G=\mathrm {SO}(2n)\). If Q is symplectic which forces \(\dim V=2n\), we say that G is of type \(C_n\) and write \(G=\mathrm {Sp}(2n)\).
3 Connections to the Littlewood–Richardson semigroup
In this section, we discuss the connection between the detection by \(\otimes ^3\) and the Littlewood–Richardson semigroup \(\mathrm {LR}_n\) of order n. We begin this section by recalling the definition of \(\mathrm {LR}_n\) (see [28] for instance).
We prove the following key proposition:
Proposition 3.1
The representation \(\otimes ^3\) detects \(\mathbb {S}_{\parallel \lambda \parallel }(G)\) if and only if \(N^{\lambda }_{\lambda \lambda } > 0\), where \(N^{\lambda }_{\lambda \lambda }\) is the common structure constant as in (2.6).
Proof
Let V be the standard representation of G. Then as described in Sect. 2, one obtains irreducible representations \(\mathbb {S}_{\parallel \lambda \parallel }(V)\) of G with highest weight \(\lambda \).
By Proposition 3.1, the detection by \(\otimes ^3\) is equal to nonvanishing of \(N^{\lambda }_{\lambda \lambda }\). Hence the Newell–Littlewood formula of (2.6) implies the following corollary:
Theorem 3.2
The subgroup \(\mathbb {S}_{\parallel \lambda \parallel }(G)\) is detected by \(\otimes ^3\) if and only if there are \(\alpha , \beta , \gamma \in P_n\) such that all triples \((\lambda , \alpha , \beta )\), \((\lambda , \alpha , \gamma )\) and \((\lambda , \beta , \gamma )\) are elements in \(\mathrm {LR}_n\). \(\square \)
Theorem 3.3
Let \(\lambda \) be a partition of an odd number. Then the representation \(\otimes ^3\) does not detect \(\mathbb {S}_{\parallel \lambda \parallel }(G)\).
Proof
One has \(c^{\lambda }_{\alpha \beta }=0\) unless \(\alpha +\beta =\lambda \). Thus \(N^{\lambda }_{\mu \nu }\) is zero unless \(\lambda +\mu +\nu \) is even. Hence \(N^{\lambda }_{\lambda \lambda }=0\) unless \(\lambda \) is even. \(\square \)
Since the Littlewood–Richardson constant \(c^{\lambda }_{\mu \nu }\) is the same as the conjugate Littlewood–Richardson constant \(c^{\lambda '}_{\mu '\nu '}\) as in (2.3), we also obtain the following result:
Theorem 3.4
Let \(\lambda \in P_n\) be such that its conjugate \(\lambda '\) is also in \(P_n\). Then \(\mathbb {S}_{\parallel \lambda \parallel }(G)\) is detected by \(\otimes ^3\) if and only if \(\mathbb {S}_{\parallel \lambda '\parallel }(G)\) is detected by \(\otimes ^3\).
Proof
Let \(\lambda \in P_n\) be a partition such that its conjugate \(\lambda '\) is in \(P_n\) as well. Suppose \(\mathbb {S}_{\parallel \lambda \parallel }(G)\) is detected by \(\otimes ^3\). Then \(N^{\lambda }_{\lambda \lambda }>0\) which implies, by Theorem 3.2, there exist \(\alpha , \beta , \gamma \in P_n\) so that \(c^{\lambda }_{\alpha \beta }>0\), \(c^{\lambda }_{\alpha \gamma }>0\) and \(c^{\lambda }_{\beta \gamma }>0\).
On the other hand, by (2.3), one knows that \(c^{\lambda '}_{\alpha '\beta '}=c^{\lambda }_{\alpha \beta }>0\), \(c^{\lambda '}_{\alpha '\gamma '}=c^{\lambda }_{\alpha \gamma }>0\) and \(c^{\lambda '}_{\beta '\gamma '}=c^{\lambda }_{\beta \gamma }>0\), which in turn implies \(N^{\lambda '}_{\lambda '\lambda '}>0\) as well. The fact \(\lambda ' \in P_n\) together with the definition of the Littlewood–Richardson coefficient forces \(\alpha ', \beta ', \gamma '\) to be elements in \(P_n\). This completes the proof. \(\square \)
4 Explicit constructions of subgroups detected by \(\otimes ^3\)
As mentioned briefly in the introduction, describing \(\mathrm {LR}_n\) explicitly is not obvious at all in general (see [28]). In this section, we will explore the combinatrical contructions on \(\lambda \) explicitly such that \(N^{\lambda }_{\lambda \lambda }>0\). The constructions are purely based on the Littlewood–Richardson rule summarized in Sect. 2.2.
We restate Theorem 1.6 for the reader’s convenience:
Theorem 4.1
 (a)
all \(\lambda _i\) are even, or
 (b)
it has even \(\ell (\lambda )\) and all nonzero \(\lambda _i\) are distinct and odd, or
 (c)
\(\lambda \) is a hook partition,
We will consider all cases separately in following subsections. Note that rectangular partition is a special case of (a).
4.1 All parts are even
In this section, we consider the first case of Theorem 4.1.
Proposition 4.2
Let \(\lambda \in P_n\) be a partition such that all parts \(\lambda _i\) are even. Then the representation \(\otimes ^3\) detects \(\mathbb {S}_{\parallel \lambda \parallel }(G)\).
Proof
The following example shows the skew tableau of shape \(\lambda \alpha \) with content \(\beta \):
Example 1
4.2 Even \(\ell (\lambda )\) and all parts are distinct and odd
In this section, we consider the partitions \(\lambda \) such that \(\ell (\lambda )\) is even and all parts are distinct and odd.
Proposition 4.3
Let \(\lambda \in P_n\) be a partition such that \(\ell (\lambda )\) even and all nonzero \(\lambda _i\) are distinct and odd. Then the representation \(\otimes ^3\) detects \(\mathbb {S}_{\parallel \lambda \parallel }(G)\).
Proof
Showing \(c^{\lambda }_{\alpha \beta }\geqslant 1\) will be similar to the proof of Proposition 4.2: For \(1\leqslant i\le k\), put i’s into \(\tfrac{\lambda _i1}{2}\) boxes in the ith row of \(\beta \) and add them to the right side of the \(\tfrac{\lambda _i+1}{2}\) empty boxes in the ith row of \(\alpha \). This becomes then the ith row of \(\lambda \) with \(\lambda _i\) boxes where the first \(\tfrac{\lambda _i+1}{2}\) boxes are empty and the remaining \(\tfrac{\lambda _i1}{2}\) boxes have i’s in them. For \(k+1\leqslant i\le 2k\), put i’s into \(\tfrac{\lambda _i+1}{2}\) boxes in the ith row of \(\beta \) and add them to the right side of the \(\tfrac{\lambda _i1}{2}\) empty boxes in the ith row of \(\alpha \). This becomes then the ith row of \(\lambda \) with \(\lambda _i\) boxes where the first \(\tfrac{\lambda _i1}{2}\) boxes are empty and the remaining \(\tfrac{\lambda _i+1}{2}\) boxes have i’s in them. If \(\lambda _{2k}>1\), then the resulting tableau is just the skew tableau of shape \(\lambda \alpha \) with content \(\beta \). If \(\lambda _{2k}=1\), then the last part \(\tfrac{\lambda _{2k}1}{2}\) of \(\alpha \) becomes zero, so \(\alpha \) will have only \(2k1\) nonzero parts. In this case, we add the last box of \(\beta \) containing 2k to the bottom of the first column of \(\alpha \) (compare with the Young diagrams in the first case of Example 2 below). This becomes then the last row of \(\lambda \). This gives us the skew tableau of shape \(\lambda \alpha \) with content \(\beta \). This implies that \(c^{\lambda }_{\alpha \beta }\geqslant 1\) in either case.
Similarly, we obtain the skewtableau of shape \(\lambda \beta \) with content \(\gamma \): For \(1\leqslant i\le k\), put i’s into \(\tfrac{\lambda _i+1}{2}\) boxes in the ith row of \(\gamma \) and add them to the right side of the \(\tfrac{\lambda _i1}{2}\) empty boxes in the ith row of \(\beta \). This becomes then the ith row of \(\lambda \) with \(\lambda _i\) boxes where the first \(\tfrac{\lambda _i1}{2}\) boxes are empty and the remaining \(\tfrac{\lambda _i+1}{2}\) boxes have i’s in them. For \(k+1\leqslant i\le 2k\), put i’s into \(\tfrac{\lambda _i1}{2}\) boxes in the ith row of \(\gamma \) and add them to the right side of the \(\tfrac{\lambda _i+1}{2}\) empty boxes in the ith row of \(\beta \). This becomes then the ith row of \(\lambda \) with \(\lambda _i\) boxes where the first \(\tfrac{\lambda _i+1}{2}\) boxes are empty and the remaining \(\tfrac{\lambda _i1}{2}\) boxes have i’s in them. If the last part \(\tfrac{\lambda _{2k}1}{2}\) of \(\gamma \) is zero, we don’t have any box to be added. In that case, the last row of \(\beta \) will be just the last row of \(\lambda \) (compare with the Young diagrams in the second case of Example 2 below). This implies that \(c^{\lambda }_{\beta \gamma }\geqslant 1\).
Now, in order to prove that \(c^{\lambda }_{\alpha \gamma }\geqslant 1\), we will need to modify the above process slightly: For each \(1\leqslant j\le 2k\), put j’s into all boxes in the jth row of \(\gamma \). To right of the first row of \(\alpha \), add only \(\tfrac{\lambda _11}{2}\) boxes with 1’s in them. This ensures \(\lambda _1\) boxes in the first row of \(\lambda \). The last box with 1 in it should be added to right to the second row of \(\alpha \). After that, we add the boxes with 2 in them in the second row until we reach to \(\lambda _2\) boxes all together. Whatever the remaining boxes with 2 should be added into the third row. We repeat this process until we add all the boxes of \(\gamma \) with numbers in them (compare with the Young diagrams in the third case of Example 2 below). In this way, we obtain the skew tableau of shape \(\lambda \alpha \) with content \(\gamma \). Therefore we prove that \(c^{\lambda }_{\alpha \gamma }\geqslant 1\) which completes the proof. \(\square \)
The following example shows how to obtain skew tableau of shape \(\lambda \alpha \) with content \(\beta \), skew tableau of shape \(\lambda \beta \) with content \(\gamma \), and skew tableau of shape \(\lambda \alpha \) with content \(\gamma \), respectively, by the process given in the proof of Proposition 4.3.
Example 2
4.3 Hook partitions
Proposition 4.4
Let \(\lambda \in P_n\) be a hook partition of an even number. Then the representation \(\otimes ^3\) detects \(\mathbb {S}_{\parallel \lambda \parallel }(G)\).
Proof
Fix \(b\in \{0, 1, 2, \dots , n1\}\) and let \(\lambda =(1+a, 1^b)\) be the partition such that \(1+a+b\) is even. Note that for \(\mathrm {SO}(2n)\), b ranges only up to \(n2\) due to the constraint on the \(\lambda _n\) being zero.
Again, one needs to construct partitions \(\alpha \), \(\beta \) and \(\gamma \) such that \(c^{\lambda }_{\alpha \beta }>0\), \(c^{\lambda }_{\alpha \gamma }>0\) and \(c^{\lambda }_{\beta \gamma }>0\). Note that in order for \(\lambda \) to be hook partition, all \(\alpha \), \(\beta \), \(\gamma \) must be hook partitions as well. Since \(\lambda =1+a+b\) is even, we have two cases to consider: one is where a is odd and b is even and the other is where a is even but b is odd.
Next, assume that a is even and b is odd. Then the conjugate partition \(\lambda '\) belongs to the case just mentioned. By Theorem 3.4, we obtain the desired result. \(\square \)
Now we provide an example to explain the proof above.
Example 3
Declarations
Open Access
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Acknowledgements
The author is grateful to L. Saper for answering various questions on representation theory and J. R. Getz for his constant support throughout this project and help with editing of the paper. The author also thanks to anonymous referees for useful comments and pointing out the Littlewood–Richardson semigroup and its saturation property.
Authors’ Affiliations
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