Plethysms of symmetric functions and representations of SL2(C)

Let ∇λ denote the Schur functor labelled by the partition λ and let E be the natural representation of SL2(C). We make a systematic study of when there is an isomorphism ∇λSymE ∼= ∇μSymE of representations of SL2(C). Generalizing earlier results of King and Manivel, we classify all such isomorphisms when λ and μ are conjugate partitions and when one of λ or μ is a rectangle. We give a complete classification when λ and μ each have at most two rows or columns or is a hook partition and a partial classification when ` = m. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when ∇λSymE is irreducible. The methods used are from representation theory and combinatorics; in particular, we make explicit the close connection with MacMahon’s enumeration of plane partitions, and prove a new q-binomial identity in this setting.


Introduction
Let SL 2 (C) be the special linear group of 2 × 2 complex matrices of determinant 1 and let E be its natural 2-dimensional representation. The irreducible complex representations of SL 2 (C) are, up to isomorphism, precisely the symmetric powers Sym n E for n ∈ N 0 . A classical result, discovered by Cayley and Sylvester in the setting of invariant theory, states that if a, b ∈ N then the representations Sym a Sym b E and Sym b Sym a E of SL 2 (C) are isomorphic. More recently, King and Manivel independently proved that ∇ (a b ) Sym b+c−1 E is invariant, up to SL 2 (C)-isomorphism, under permutation of a, b and c. Here ∇ (a b ) is an instance of the Schur functor ∇ λ , defined in § 2.4. Motivated by these results, the purpose of this article is to make a systematic study of when there is a plethystic isomorphism (1) ∇ λ Sym E ∼ = ∇ µ Sym m E of SL 2 (C)-representations. By taking the characters of each side, (1) is equivalent to (2) q − |λ|/2 s λ (1, q, . . . , q ) = q −m|µ|/2 s µ (1, q, . . . , q m ).
where s λ is the Schur function for the partition λ. By Remark 2.9, we also have s λ (1, q, . . . , q ) = (s λ • s ) (1, q) where • is the plethysm product. Thus (1) can be investigated using a circle of powerful combinatorial ideas; these include Stanley's Hook Content Formula [27,Theorem 7.12.2]. Our results reveal numerous surprising isomorphisms, not predicted by any existing results in the literature, and a number of new obstacles to plethystic isomorphism.
In particular, we prove a converse to the King and Manivel result. We also note Lemma 4.4, which implies that, in the typical case for (2), the Young diagrams of λ and µ have the same number of removable boxes. Borrowing from the title of [15], these are all cases where one may "hear the shape of a partition".
Main results. Let (λ) denote the number of parts of a partition λ and let a(λ) denote its first part, setting a(∅) = 0. We refer to the relation ∼ m as plethystic equivalence. By Lemma 4.1, we have λ ∼ (λ)−1 (λ)−1 λ, where, by definition, λ is λ with its columns of length (λ) removed. Such plethystic equivalences arise from the triviality of the representation +1 Sym of SL(E); as we show in Example 1.12 below, they can be dispensed with by using Lemma 4.2 and Proposition 4.3 to reduce to the following "prime" case. To avoid technicalities, we state our first main theorem in a slightly weaker form than in the main text. Given a partition λ with Durfee square of size d, let EP(λ) be the partition, shown in Figure 1 in § 5, obtained from the first d rows of λ by deleting the maximal rectangle containing the Durfee square of λ. Let SP(λ) be defined analogously, replacing rows with columns. Thus SP(λ) = EP(λ ) where λ is the conjugate partition to λ. The "if" direction of the theorem below was proved by King in [16, § 4] and is also Cagliero and Penazzi's main result in [3]. Our second main theorem sharpens this result to show that "infinitely many" may be replaced with "three", and, in the case of prime equivalences, with "two". (iii) There exist distinct n, n † ∈ N such that λ ∼ n n µ and λ ∼ n † n † µ if and only if λ = µ.
It is clear that no still sharper result can hold in (i) or (iii); Example 1.12 below shows that the same is true for (ii).
If (λ) r, let λ •r denote the complementary partition to λ in the r × a(λ) box, defined formally by λ •r r+1−i = a(λ) − λ i for 1 i r. The "if" direction of the following theorem was proved in [16, § 4]. Our fourth main result includes the converse of the King and Manivel six-fold symmetries mentioned at the outset. Again, to avoid technicalities, we state it in a slightly weaker form below.
Algebraic Combinatorics, Vol. 4 #1 (2021) Since ∇ λ Sym E is irreducible if and only if λ ∼ 1 (m) for some m ∈ N 0 , Corollary 1.8 can also be obtained from the full version of Theorem 1.6, or, more directly, from Corollary 1.7.
In § 10 we classify all equivalences λ ∼ m µ when λ and µ are two-row, two-column or hook partitions. To give a good flavour of this, we state the result for equivalences between two-row and hook partitions. Theorem 1.9. Let λ be a non-hook partition with exactly two parts and let µ be a hook partition with non-zero arm length and leg length. If (λ) and m (µ) then λ ∼ m µ if and only if the relation is one of In § 11 we consider the case of prime equivalences in which = m. Building on Theorem 1.5, we obtain the following partial classification.  5 there exist infinitely many distinct pairs (λ, µ) such that λ = µ, λ = µ •( +1) , and λ ∼ µ. Algebraic Combinatorics, Vol. 4 #1 (2021) We end in § 12 where we show that there exist infinitely many partitions whose only plethystic equivalences are the inevitable column removals from Lemma 4.1 and the complement equivalences from Theorem 1.5.
The complex behaviour of plethystic equivalences revealed by our main theorems strongly suggests that a complete classification is infeasible. By size of the partitions, the smallest example of a plethystic equivalence not explained by any of our results is (3, 3, 2) ∼ 9 10 (2, 2, 2, 2, 1, 1). The following example is chosen to illustrate many of our main results.
By Lemma 4.2 this chain can be read as the factorization of a non-prime plethystic equivalence By Theorem 1.4(ii), there are precisely two plethystic equivalences between (b + c) b , b d and (b+c) c , c d , namely the two found above. As expected from this theorem, only one of these equivalences is prime.
1.1. Outline. In the remainder of this introduction we illustrate the critical Theorem 3.4, which collects a number of equivalent conditions for the plethystic equivalence in Definition 1.1 by giving two short proofs that Sym a Sym b E ∼ = Sym b Sym a E for all a, b ∈ N 0 . In the spirit of this work, one proof also gives a converse. We then give a brief literature survey, organized around the different generalizations of this isomorphism.
In § 2 we construct the irreducible representations of SL 2 (C) as the symmetric powers Sym E, and give other basic results from representation theory. We then give an explicit model for the representations ∇ λ Sym E. While ∇ λ E is non-zero only if (λ) 2, the representation ∇ λ Sym E is non-zero whenever (λ) − 1. This explains the ubiquity of this condition in this work, and why we require the generality of Schur functors, despite working only with representations of GL 2 (C) and its subgroups. To make the paper largely self-contained, we end by defining Schur functions.
The reader may prefer to treat § 2 as a reference and begin reading in § 3, where we state and prove Theorem 3.4. In § 4 we collect some useful basic properties of the relations ∼ m . In § § 5-12 we prove the main results, as already outlined. Theorem 1.5 requires the statement of Theorem 1.4, which in turn uses the statement of Theorem 1.3; several later theorems need the statement of Theorem 1.5. Apart from this, the sections may be read independently.  [4, § 20], where he acknowledges Hermite's prior discovery; some special cases may be seen in Sylvester [29], published in the same journal issue as [13]. Thus it is also known (for instance in the title of [20]) as the Cayley-Sylvester formula. An invariant theory proof in modern language may be found in [26, 3.3.4]. Another elegant proof, using the symmetric group, is in [12,Corollary 2.12]. We offer two proofs that illustrate different conditions in Theorem 3.4. Each shows that (n) ∼ n ( ), or equivalently, Hermite reciprocity for representations of SL 2 (C). Then, since the degrees on each side are equal, it follows from Proposition 3.6 that there is a GL 2 (C)-isomorphism. The first proof is well known, and is sketched in [11,Exercise 6.19]; later in § 8.1 we give its generalization to plane partitions. Yet another proof (including the converse) can be given using Theorem 3.4(i).
Proof by tableaux. By Theorem 3.4(g), we have (n) ∼ n ( ) if and only if |S e n | = |S n e | for all e ∈ N 0 , where, by definition, S e (n) is the set of semistandard tableaux of shape (n) with entries from {0, 1, . . . , } whose sum of entries is e. Let t be such a tableau, having exactly c i entries of i for each i ∈ {0, 1, . . . , b}. Then, reading its unique row from right to left, and ignoring any zeros, t encodes the partition ( c , . . . , 1 c1 ) of e.
Hence |S e (n)| is the number of partitions of e whose diagram is contained in the n × box. By conjugating partitions, this number is invariant under swapping n and . where / denotes a difference multiset, as defined in § 3.1. Equivalently, the multiset unions { + 1, + 2, . . . , n + } ∪ {1, 2, . . . , n } and {m + 1, m + 2, . . . , m + n } ∪ {1, 2, . . . , n} are equal. If n = n then, cancelling {1, 2, . . . , n} from each side, we see that = m, giving a trivial solution. Otherwise we may suppose by symmetry that n < n . Now n + 1 is in the first union and so m = n; comparing greatest elements we see that n = . We therefore have n = and m = n, corresponding to Hermite reciprocity.
We remark that the first proof shows that that partitions contained in the n × box are enumerated, according to their size, by a character of SL 2 (C). In particular by Theorem 3.4, the sequence is unimodal that is, first weakly increasing and then weakly decreasing.
1.3. Literature on SL 2 (C)-plethysms. By Hermite reciprocity, the multiplicity of any Schur function s ( n−d,d) labelled by a two-part partition is the same in s (n) •s ( ) and s ( ) • s (n) . More generally, Foulkes conjectured in [8] that if n then s (n) • s ( ) − s ( ) • s (n) is a non-negative integral linear combination of Schur functions; Foulkes' Conjecture has been proved only when n 5 (see [5]) and when n is very large compared to (see [2]). A number of "stability" results on plethysm are relevant to this setting. For example, a special case of the theorem on page 354 of [2] implies that the multiplicity of Sym r E in Sym n Sym E is at most the multiplicity of Sym r+n E in Sym n Sym +1 E. The first proof of Hermite reciprocity above translates this into a non-trivial combinatorial result comparing partitions of r in the n × box and partitions of r + n in the n × ( + 1) box.

Rowena Paget & Mark Wildon
In [16], King proves the "if" direction of Theorem 1.6, and sketches a proof of a weaker version of the converse. He mentions as one motivation the Wronskian isomorphism b Sym b+c−1 E ∼ = Sym b Sym c E of representations of the compact subgroup SU 2 (C) of SL 2 (C). This is interpreted by Wybourne in [31] as an equality between the number of completely antisymmetric states of b + c − 1 identical bosons each of angular momentum c/2 and the number of symmetric states of b identical bosons each of angular momentum c/2. (There is a typographic error between (13) and (14) in [31]; m+1+n should be m+1−n, as in (13).) This realizes the well-known equality between the number of c-multisubsets of {1, . . . , b} and the number of c-subsets of {1, . . . , b+c−1}. By Lemma 5.1 in [21], the special case of the Wronskian isomorphism 2 Sym c+1 E ∼ = Sym 2 Sym c E holds when E is the natural representation of any finite special linear group SL 2 (F q ) when q is odd. It would be interesting to have further examples of such "modular plethysms".
The second main result of [1] classifies all partitions λ and ν such that the plethysm s λ • s ν is equal to a single Schur function. Apart from the obvious s λ • s (1) = s λ , the only examples are s (1,1) • s (1,1) = s (2,1,1) and s (1,1) • s (2) = s (3,1) . By Remark 2.9 and (10), the formal character of ∇ λ Sym E, evaluated at 1 and q is (s λ • s ( ) )(1, q). Our Corollary 1.8 therefore shows that there are more irreducible plethysms when we work with symmetric functions truncated to two variables. The equality (s (1 5 gives a similar "non-generic" example for three variables. Corollary 1.8 is itself a special case of Theorem 9.5 on skew Schur functors. While we do not require it in this work, we note that a combinatorial formula for the corresponding plethysm s λ/λ (1, q, . . . , q ) = (s λ/λ • s ( ) )(1, q) is given by Morales, Pak and Panova in [22,Theorem 1.4] in terms of certain "excited" Young diagrams of shape λ/λ first defined by Ikeda and Naruse in [14]. This result is a generalization of Stanley's Hook Content Formula (see [27,Theorem 7.21.2]), one of the main tools in this work. As a corollary the authors obtain a formula due to Naruse [23] for the number of standard tableaux of shape λ/λ .
For further general background on plethysms we refer the reader to [17] and to the survey in [24].
2. Background 2.1. Representations of SL 2 (C). Let G be a subgroup of GL 2 (C) containing SL 2 (C). A representation ρ : G → GL(V ) is said to be polynomial if, with respect to a chosen basis of V , each matrix entry in ρ(g) is a polynomial in the matrix entries of g ∈ G. If these polynomials all have the same degree r, we say that V has degree r. We define the character of a polynomial representation V of G to be the unique two variable polynomial Φ V such that for all non-zero α, β ∈ C. We define the Q-character of V to be the Laurent polynomial Remarkably every smooth representation of SL 2 (C) is polynomial. Thus the following summary theorem is a restatement of a basic result in Lie Theory.
Let E be a 2-dimensional complex vector space with basis e 1 , e 2 . The diagonal matrix with entries 1/α and α acts on the canonical basis element e −k 1 e k 2 of Sym E by multiplication by α 2k− . Therefore and so Proof. Since the Laurent polynomials in (4) are linearly independent, the result is immediate from Theorem 2.1.

2.2.
Partitions. Let Par(r) denote the set of partitions of r ∈ N. We write |λ| = r if λ ∈ Par(r). We have already introduced the notation (λ) for the number of parts of λ. If i > (λ) then we set λ i = 0. The Young diagram of λ is the set {(i, j) : 1 i (λ), 1 j λ i }; we refer to its elements as boxes, and draw [λ] using the "English" convention with its longest row at the top of the page, as in Example 2.6 below.
, then we say that t has entry b in box (i, j), and write t (i,j) = b. If the entries of each row of t are weakly increasing when read from left to right we say that t is row-semistandard. If the entries of each column of t are strictly increasing when read from top to bottom, we say that t is column standard. If both conditions hold, we say that t is semistandard. Let RSSYT (λ) and SSYT (λ) be the sets of row semistandard and semistandard λ-tableaux respectively, with entries in {0, 1, . . . , }. Note that 0 is permitted as an entry. Given a permutation σ of the boxes [λ], and a λ-tableau t, we define σ · t by (σ · t)(i, j) = t σ −1 (i, j) . Thus if t has entry b in box (i, j) then σ · t has entry b in box σ(i, j). Let C(λ) be the group of all permutations that permute within themselves boxes in the same column of [λ]. We define the weight of tableau t, denoted |t|, to be the sum of its entries.

2.4.
A construction of ∇ λ Sym E. Fix ∈ N and let V = v 0 , . . . , v be an ( + 1)-dimensional complex vector space. Given a λ-tableau t with entries from {0, 1, . . . , }, define We say that F (t) is the GL-polytabloid corresponding to t. Observe that if σ ∈ C(t) then Hence F (t) = 0 if t has a repeated entry in a column. It is clear that {f (t) : t ∈ RSSYT (λ)} is a basis of (λ) i=1 Sym λi V . Thus given any g ∈ GL(V ), there exist unique coefficients α s ∈ C for s ∈ RSSYT (λ) such that Algebraic Combinatorics, Vol. 4 #1 (2021) It is routine to check that if σ is a permutation of [λ] then It now follows from the definition in (6) that the linear span of the F (t) for t a λtableau with entries from {0, 1, . . . , } is a GL(V )-subrepresentation of Example 2.3. By (7) the representation ∇ (1 n ) V has as a basis all GL-polytabloids F (t) where t is a standard (1 n )-tableau with entries from {0, 1, . . . , }. Moreover, the linear map More generally we have the following theorem. Let v k = e −k 1 e k 2 for 0 k be the canonical basis of Sym E. Using this basis in Definition 2.5, the action of g ∈ GL(E) on a GL-polytabloid F (s) may be computed by the following device: formally replace each entry b of s with gv b , expressed as a linear combination of v 0 , v 1 , . . . , v . Then expand multilinearly, and use the column relation (7) followed by Garnir relations (see [7,Corollary 2.6] or [10,Chapter 8]) to express the result as a linear combination of GL-polytabloids F (t) for semistandard tableaux t.
Example 2.6. Take = 2 so V = Sym 2 E = e 2 1 , e 1 e 2 , e 2 2 . The action of a lowertriangular matrix g ∈ GL 2 (C) on V is given, with respect to the chosen basis, by In its action on ∇ (2,1) Sym 2 E we have where the third line uses the column relation in (7).
Lemma 2.7. Let λ be a partition and let ∈ N 0 . We have Algebraic Combinatorics, Vol. 4 #1 (2021) Proof. By Theorem 2.4, the F (t) for t ∈ SSYT (λ) are a basis of ∇ λ Sym E. Let g ∈ GL 2 (C) be diagonal with entries α and δ. Let τ ∈ C(t) and let u = τ · t. By (5), is the weight |u| defined above. This is also the weight of t. Therefore each F (t) is an eigenvector for g with eigenvalue α |λ|−|t| δ |t| . The lemma follows.
Definition 2.8. Let λ be a partition. Given a λ-tableau t with entries from N 0 , let is the number of entries of t equal to k ∈ N 0 . The Schur function s λ is the symmetric function defined by The compatibility condition (8) is easily checked. Let C[q] be a polynomial ring. Observe that when x k is specialized to q k , the monomial x t becomes q |t| , where, as usual, |t| is the weight of t. Therefore It follows immediately from our definition and Lemma 2.7 that This equation is the main bridge we need between representation theory and combinatorics.
For Theorem 3.4(i) we require the original definition of Schur polynomials using determinants and antisymmetric polynomials. Given a sequence (γ 0 , γ 1 , . . . , γ ) of non-negative integers, define (11) a γ (x 0 , x 1 , . . . , is the weight of the semistandard λ-tableau having λ i entries of i − 1 in row i; as the terminology suggests, this tableau has the minimum weight of any tableau in SSYT (λ). It follows that q b(λ) is the summand of s λ (1, q, . . . , q ) of minimum degree.
For example, the unique greatest hook length of a non-empty partition λ is h (1,1)

Theorem 2.12 (Stanley's Hook Content Formula). Let λ be a partition and let ∈ N. Then
Proof. This is a restatement of [27, Theorem 7.21.2] using the quantum integer notation. Note that our appears in [27] as − 1.

2.7.
Pyramids. In this subsection we prove an antisymmetric analogue of Stanley's Hook Content Formula. Most of the ideas may be found in [27, § 7.21], so no originality is claimed.

Plethysms of symmetric functions
By (11) we have Therefore, specializing x j to q γj in the Vandermonde determinant, we obtain Set γ j = λ j+1 + − j for 0 j , and use the observation just before the lemma to get a λ+( ,..., Taking the ratio of these two equations and using (12) we obtain . This is equivalent to the claimed identity.
Proof. By Lemma 2.14 there exists c ∈ N 0 such that The factors in the numerator are the quantum integers from ∆ (λ). The factors in the denominator are the quantum integers from {1 , 2 −1 , . . . , }, where the exponents indicate multiplicities. By (9), q b(λ) is the monomial of least degree in s λ (1, q, . . . , q ). Since each quantum integer is congruent to 1 modulo q, the result follows.

Equivalent conditions for the plethystic isomorphism
3.1. Difference multisets. The following formalism simplifies the main results of this section and is convenient throughout this paper.
Alternatively, a difference multiset may be regarded as an element of the free abelian group on N. This point of view justifies our definition of equality and makes obvious many simple algebraic rules for manipulating difference multisets. For example, The following lemma is used implicitly in (4.8) in [16].
Proof. If either X or Y is empty the result is obvious. In the remaining cases, let u be greatest such that x∈X (q x − 1) has e 2πi/u as a root. By choice of u, q u − 1 is a factor in the left-hand side. Since e 2πi/u is also a root of y∈Y (q y − 1), the same argument shows that q u − 1 is a factor in the right-hand side. Therefore u = max X = max Y and it follows inductively that X = Y .
Proof. Multiply through by z∈Z (q z − 1) w∈W (q w − 1) and then apply Lemma 3.2.
We apply this corollary to the polynomial quotients in Theorem 2.12 and Corollary 2.15 in the proof of Theorem 3.4 below.

Portmanteau Theorem.
Recall from § 2.3 that the weight of a tableau, denoted |t|, is its sum of entries. The minimum weight b(λ) is defined in Definition 2.10. Given a partition λ and ∈ N 0 , let S e (λ) be the set of all semistandard λ-tableaux with entries from {0, 1, . . . , } whose weight is e. Thus (9) can be restated as (14) s In (h) and (i) below the notation indicates a difference multiset, as defined in the previous subsection. The multisets C(λ) and H(λ) are defined in Definition 2.11 and ∆ (λ) is defined in Definition 2.13.
Hence it is centrally symmetric and unimodal about its constant term Now, comparing points of central symmetry, We noted before Definition 2.11 that s λ (1, q, . . . , q ) has minimum degree summand q b(λ) . Therefore (e) and (f) are equivalent. By (14), (e) and (g) are equivalent. The remainder of the "moreover" part now follows by comparing the q powers in (d) and (f). By Stanley's Hook Content Formula, as stated in Theorem 2.12, (f) holds if and only if By Corollary 3.3, this is equivalent to (h). Finally (f) and (i) are equivalent by the same argument with Corollary 3.3 applied to the right-hand side in Corollary 2.15.
3.3. Extending plethystic isomorphisms. We end by considering when an The following lemma is used to show that the only obstruction is the determinant representation of GL(E).
Lemma 3.5. Let λ and µ be partitions and let , m ∈ N be such that Proof. By the "moreover" part in Theorem 3.4, Since representations of GL 2 (C) are completely reducible, they are determined by their characters. Since Φ det (α, β) = αβ, the GL 2 (C) representations det D ⊗∇ λ Sym E and to ∇ µ Sym m E are isomorphic if and only if for all α, β ∈ C × . Using (15) to rewrite Φ ∇ µ Sym m E (1, β/α) on the right-hand side we get the equivalent condition that (αβ) D α |λ| = α m|µ| (β/α) D for all α, β ∈ C × . This holds by our choice of D.
G GL 2 (C) then either {det g : g ∈ G} is one of the subgroups U d or it is dense (in the Zariski topology) on C × . In the latter case, if V and W are polynomial representations of GL 2 (C) and V ∼ = W as representations of SL 2 (C), then the isomorphism extends to G if and only if it extends to GL 2 (C).
Proposition 3.6. Let λ and µ be partitions and let , m ∈ N be such that Proof. By Lemma 3.5, we have the required isomorphism if and only if det D is the trivial representation of MGL In particular, ∇ λ Sym E and ∇ µ Sym m E are isomorphic as representations of GL 2 (C) if and only if they are isomorphic as representations of SL 2 (C) and the degrees |λ| and m|µ| are equal. This is Theorem 3.1(ii) in [3], which our Proposition 3.6 extends.
It is worth noting that the proof of Lemma 3.5 made essential use of the fact that the representations involved are homogeneous. For example, if V = det ⊕ det 2 and W = E ⊗ det then Φ V (1, q) = Φ W (1, q) = q + q 2 . But V and W are not isomorphic, even after restriction to SL 2 (C).

Basic properties of equivalence
Given a non-empty partition λ let λ be the partition obtained by removing all columns of length (λ) from λ.
Proof. Let = (λ) − 1 and suppose that λ has precisely c columns of length (λ). We may suppose that a(λ) > c. If t is a semistandard tableau of shape λ with entries from {0, 1, . . . , } then the first c columns of t each have entries 0, 1, . . . , read from top to bottom. Let t be the tableau obtained from t by removing these columns. Using the model for ∇ λ Sym E in Definition 2.5, we see that there is a linear If g ∈ GL(E), each matrix coefficient of g in its action on Sym E is a polynomial of degree , and so the determinant of g acting on Sym E has degree ( + 1). We deduce that φ is an isomorphism of representations of GL(E). Hence by Theorem 3.4(b), we have λ ∼ λ, as required.
An alternative, but we believe less conceptual, proof of Lemma 4.1 can be given by applying Theorem 3.4(g) to the bijection t → t. When = m the relation ∼ m is neither reflexive nor transitive. The following lemma is the correct replacement for transitivity.
. For each removable box in λ , there is a corresponding semistandard tableau of shape λ and weight b(λ ) + 1, obtained from the unique semistandard tableau of shape λ and minimal weight b(λ ) by increasing the entry in the removable box by 1. Conversely, every element of S b(λ ) (λ ) arises in this way. A similar result holds for µ . The displayed equation therefore implies that the numbers of removable boxes are the same.
Recall that a(λ) denotes the first part of a partition λ.
Lemma 4.6. Let λ be a partition and let ∈ N be such that (λ). The unique greatest element of C(λ) + + 1 is a(λ) + .
Proof. The box 1, a(λ) of [λ] has the unique greatest content of any box in λ, namely of a(λ) − 1.
The rank of a non-empty partition λ, denoted R(λ), is the maximum r such that λ r r. The Durfee square of λ is the subset {(i, j) : 1 i, j R(λ)} of its Young diagram. Theorem 1.3 also requires the following less standard definitions.
. . . We begin with three equivalent conditions for the existence of infinitely many plethystic equivalences between distinct partitions λ and µ. We then prove a fourth equivalent condition, namely that µ = λ and EP(λ) = SP(λ) , obtaining Theorem 5.5 and, a fortiori, Theorem 1.3.
Proposition 5.2. Let λ and µ be non-empty partitions. The following are equivalent.
Work of Craven [6] shows that there is no simple characterization of when H(λ) = H(µ). Fortunately the second condition in (iii) is much more tractable.  Proof. The "if" direction is implied by the special case when E(µ) = E(λ) + 1 and S(µ) = S(λ) − 1. In this case µ is obtained from λ by deleting the lowest of the S(λ) parts of λ of size R(λ) and inserting R(λ) boxes as a new column at the right of the E(λ) columns of λ of size R(λ). We must show that C(µ) = C(λ) + 1. It is clear that adding 1 to the content of the boxes (i, j) ∈ [λ] with i > R(λ) + S(λ) or j > R(λ) + E(λ) gives the content of a corresponding box (i − 1, j) or (i, j + 1) ∈ [µ]. Moreover, as the shaded squares in Figure 2 show in a special case, adding 1 to the content of each remaining box in [λ] gives the contents of the remaining boxes in [µ]. Conversely, suppose that C(λ) + d = C(µ). It is clear that no member of C(λ) can have multiplicity exceeding R(λ). As can be seen from the content of the two marked boxes in Figure 1, the contents of multiplicity R(λ) are precisely − S(λ), . . . , E(λ). Similarly in C(µ) the contents of maximum multiplicity are − S(µ), . . . , E(µ), each with multiplicity R(µ). Therefore R(λ) = R(µ), E(λ) + d = E(µ) and − S(λ) + d = − S(µ).
where the parentheses indicate the arm and leg lengths. Similarly if (i, j) is in a box in the Durfee square of µ then Hence h (i,j) (λ) = h (i,j) (µ). It remains to compare the hook lengths

Plethysms of symmetric functions
The unique greatest elements of each side are ν 1 −2 and κ 1 −2, respectively. Therefore ν 1 = κ 1 . Cancelling the equal sets from each side we may repeat this argument inductively, as in the proof of Lemma 5.3, to get ν = κ, as required.
We are now ready to prove the slightly stronger version of Theorem 1.3 stated below. We end this section by remarking that, by Proposition 3.6, a plethystic isomorphism ∇ λ Sym (λ)−1+k E ∼ = ∇ λ Sym a(λ)−1+k E given by Theorem 5.5 extends to the overgroup MGL

Multiple equivalences
We need two lemmas on difference multisets. the minimum element in the left-hand side with non-zero multiplicity is min X, with negative multiplicity, and so min X = min Y + c. Lemma 6.2. Let Z and W be finite multisubsets of Z and let t ∈ Z be non-zero. If Proof. Suppose, for a contradiction, that Z = W . Let x be the greatest element with non-zero multiplicity in Z/W . Clearly x + t is the greatest element with non-zero multiplicity in (Z + t)/(W + t). But Z/W = (Z + t)/(W + t) so x = x + t, hence t = 0, a contradiction.
Proof of Theorem 1.4(i) and (ii). If = † then from λ ∼ m µ and λ ∼ † m † µ we get µ ∼ m λ and λ ∼ m † µ, and so by Lemma 4.2, µ ∼ m m † µ. But now, by Lemma 4.5, we have m = m † , contradicting that the pairs ( , m) and ( † , m † ) are distinct. Therefore we may suppose that < † , and in (ii) of the three plethystic equivalences, two are prime. It therefore suffices to prove (i).

Rectangular equivalences and q-binomial identities
In this section we determine all plethystic equivalences λ ∼ m µ in which one or both of λ and µ is a rectangle, of the form (a b ) with a, b ∈ N. Our main result is as follows.  Let λ be a partition and let a, b, c ∈ N. Then λ ∼ b+c−1 (a b ) if and  only if λ is obtained by adding columns of length + 1 to a rectangle (a b ) with b  and (a , b , − b + 1) is a permutation of (a, b, c).
Clearly this implies Theorem 1.6. Conversely, as seen in Example 1.12, by using Lemma 4.1 and Lemma 4.2 one may reduce to the case when (λ) of a prime plethystic equivalence. Therefore Theorem 8.1 follows routinely from Theorem 1.6.
In the following subsection we use Theorem 3.4(e) to show that the "if" direction of Theorem 1.6 is the representation-theoretic realization of the six-fold symmetry group Algebraic Combinatorics, Vol. 4 #1 (2021) of plane partitions. Next we prove a new determinantal formula using q-binomial coefficients of MacMahon's generating function of plane partitions. We then prove the "only if" direction of Theorem 1.6 using certain unimodal graphs to keep track of the contents of rectangles. The section ends with the corollary for the case b = 1; this generalizes the Hermite reciprocity seen in § 1.2.
8.1. Plane partitions. Recall that a plane partition of shape λ is a λ-tableau with entries from N whose rows and columns are weakly decreasing, when read left to right and top to bottom. Let PP(a, b, c) denote the set of plane partitions π with entries in {1, . . . , c} whose shape sh(π) is contained in [(a b )]. Assigning 0 to each box of [(a b )] \ sh(π) defines a bijection between PP(a, b, c) and the set of (a b )-tableaux with entries from N 0 and weakly decreasing rows and columns. Observe that if t is such a tableau then rotating t by a half-turn and adding j − 1 to every entry in row j gives a semistandard tableau of shape (a b ) with entries in {0, 1, . . . , b + c − 1}. Again this map is bijective. Hence we have where, extending our usual notation, |π| denote the sum of entries of a plane partition π.
Proof of "if" direction of Theorem 1.6. Representing elements of PP(a, b, c) plane partitions by three-dimensional Young diagrams contained in the a × b × c cuboid, it is clear that the right-hand side of (18) This makes the invariance of (18) under permutation of a, b and c algebraically obvious. For a modern proof using (18) and Stanley's Hook Content Formula see (7.109) and (7.111) in [27]. In this section we prove Corollary 8.4, which gives a new q-binomial form for the right-hand side of MacMahon's formula. Specializing q to 1 in this corollary we obtain the attractive identity Proving the invariance of the right-hand side under permutation of a, b and c was asked as a MathOverflow question by T. Amdeberhan (1) in 2019.

Hermite reciprocity and q-binomial coefficients. Recall from (13) that [m]
As motivation, we note that, by Stanley's Hook Content Formula (Theorem 2.12), we have s (n) (1, q, . . . , q ) = n+ . We saw in the first proof of Hermite reciprocity in § 1.2 that s (n) (1, q, . . . , q ) is the generating function for partitions contained in the n ×  1, q, . . . , q m−1 ) = q ( 2 ) m . It will be convenient to denote the right-hand side by m . Using this notation, the dual Jacobi-Trudi formula (see [27,Corollary 7.16.2]) implies that A determinantal form of MacMahon's identity. We now apply row and column operations to the matrix in (20) using the following two versions of the Chu-Vandermonde identity for our scaled q-binomial coefficients. To make the article self-contained we include bijective proofs using that m is the generating function enumerating -subsets of {0, . . . , m − 1} by their sum of entries. (This easily follows from [28, Proposition 1.7.3].) A different proof of (21) using the q-binomial theorem is given in the solution to Exercise 100 in [28].
Proof. For (21) where each determinant is of an a × a matrix with entries defined by taking 0 i, j a − 1, and A = a Proof. Let M be the matrix with entries b+c b+j−i for 0 i, j a − 1 appearing in (20). Let C j denote the jth column of M , where columns are numbered from 0 up to a − 1. Let M be the matrix obtained from M by replacing C j with the linear combination Therefore M has entries b+c+j b+j−i as required for the first equality. Let R i denote the ith row of M . Let M be the matrix obtained from M by replacing R i with the linear combination Multiplying through by q −ij , we see that M has entries q −ij b+c+i+j b+j . The final equality follows.
We note that the q → 1 limit of the first formula in Proposition 8.3 is equivalent, by standard bijections between plane partitions and rhombal tilings, to (2.4) in [9].
Proof. Let N be the a × a matrix on the right-hand side. Since By direct calculation one finds that The identity now follows using 1 2 + · · · + (a − 1) 2 = a(a − 1 2 )(a − 1)/3.

Plethystic equivalences between rectangles.
In this subsection we prove the "only if" direction of Theorem 1.6. The following lemma gives one useful reduction. As seen here, it is most convenient to work with the shift applied to the content multiset: thus d = +1 in the usual notation. Recall from Definition 2.11 that h (i,j) (λ) denotes the hook length of the box (i, j) ∈ [λ].
Definition 8.6. Let λ be a non-empty partition and let d (λ). Define c (d) . The equalities on the right-hand side of (23) can easily be classified from the graphs of the content-hook functions. We include full details to save the reader case-by-case checking. As a visual guide, in inequalities and graphs we write x-coordinates relevant to the content in bold.
If b a then the graphs of c . It is easily seen that, between each adjacent pair of points, the graph is linear. The graph of h (a b ) can be found similarly.
By Lemma 8.5, we may reduce to the case where 2b d, which we now consider.
We must consider how these chains interleave. By our reduction b −b + d.
Suppose that b a. By the reduction, a + b a − b + d, and so the interleaved chain ends a − b + d < a + b. If d a + b then either d a, giving (i), or a < d  a + b, giving (ii). Otherwise a + b < d and so a < Suppose that a < b. By the reduction, a + b  Figure 4 overleaf. In the non-generic cases (i) and (ii) agree when a = d; (iii) and (iv) agree when −b+d = a+b; (v) and (vi) agree when −b+d = a+b. Therefore we need only compare cases (i), (iii) and (v) using Lemma 8.9. If (i) and (iii) agree then d = a = −b + d = a + b, hence b = 0, a contradiction. If (i) and (v) agree then d = a = −b + d = a + b, hence b = 0, again a contradiction. It is impossible for (iii) and (iv) to agree because b a in (iii) and a < b in (v).
8.4. One-row partitions. The special case of Theorem 8.1 for plethystic equivalences with a one-row partition is a natural generalization of Hermite reciprocity. It was stated as Corollary 1.7 in the introduction.
Proof of Corollary 1.7. By definition, there is an isomorphism ∇ λ Sym E ∼ = Sym a Sym c E of SL 2 (C)-representations if and only if λ ∼ c (a). By Theorem 8.1, this holds if and only if λ is obtained by adding columns of length +1 to a rectangle (a b ) and (a , b , c ) is a permutation of (a, b, c). After the usual reduction using Lemma 4.1 and Lemma 4.2, we may assume the plethystic equivalence is (a b ) ∼ c (a). Thus by Theorem 8.1, (a , b , − b + 1) is a permutation of (a, 1, c). Considering rectangles in conjugate pairs, we see that (a b ) is one of (a), ( Definition 9.1. Let ∈ N 0 and let λ/λ be a skew-partition. We say that s λ/λ is -irreducible if there exists b ∈ N 0 and m ∈ N 0 such that s λ/λ (1, q, . . . , q ) = q b (1 + q + · · · + q m ).
In (26) we show that b and m are determined in a simple way by λ/λ and . This result and Lemma 9.7 can also be proved using the following remark; it is not logically essential, but should help to motivate Definition 9.1.

Algebraic Combinatorics, Vol. 4 #1 (2021)
By a generalization of the equivalence of (a) and (e) in Theorem 3.4, s λ/λ isirreducible in the sense of Definition 9.1 if and only if the polynomial representation ∇ λ/λ Sym E of SL 2 (C) is irreducible.
Note that = 0 is permitted in Definition 9.1 and in the previous remark. Since s λ/λ (1, q, . . . , q ) is non-zero if and only if every column of [λ/λ ] has length at most + 1, the 0-irreducible skew-partitions are precisely those with at most one box in each column. 9.1. Irreducible skew Schur functions. In this section we state a classification of all skew partitions λ/λ and ∈ N such that s λ/λ is -irreducible. We then deduce Corollary 1.8. The following definition leads to a useful reduction. Definition 9.3. We say that a skew partition λ/λ is proper if λ 1 > λ 1 and λ 1 > λ 1 .
Definition 9.4. Given a proper skew partition λ/λ with a(λ) = p, we define the column lengths c(λ/λ ) ∈ N p by c(λ/λ ) j = λ j − λ j for 1 j p. We say that λ/λ is The Young diagrams of a skew 0-rectangle, a skew 1-near rectangle of width 3 and a skew 2-near rectangle are shown below; the final diagram fails the second displayed condition in (b), so is not a skew 1-near rectangle.
We can now state the classification.
Theorem 9.5. Let λ/λ be a proper skew partition. Then s λ/λ is -irreducible if and only if λ/λ is a skew -rectangle or λ/λ is a skew -near rectangle.
We immediately deduce the corollary for Schur functors labelled by partitions stated in the introduction.
Proof of Corollary 1.8. By Lemma 2.7, ∇ λ Sym E ∼ = Sym m E for some m ∈ N 0 if and only if s λ is -irreducible. The only skew 0-rectangles are one-part partitions. If, as in (b) in Definition 9.4, = 1 and so λ is either a skew 1-rectangle, in which case λ = (n/2, n/2) for some even n and m = 0, or a skew 1-near rectangle, in which case λ = (n − k, k), for some 1 k n/2 and (λ) = 2 and m = n − 2k. This gives case (i) of the corollary. If, as in (c), 2, then λ is either a skew -rectangle, of the form (p +1 ), or a skew -near rectangle; then all but the final column of λ has length + 1 and the final column (which may be the only column) has length either 1 or . By Lemma 9.11, in the first case s λ (1, q, . . . , q ) = q n/2 and m = 0; in the second case, is the minimum weight defined in Definition 2.10, and so m = . This gives case (ii) in Corollary 1.8.
To prove Theorem 9.5 we need the preliminary results in the following subsection. 9.2. Unimodality of specialized skew Schur functions. Fix a skew partition λ/λ of size n. The minimum weight defined in Definition 2.10 generalizes as follows to skew tableaux. Definition 9.6. We define the minimum weight of λ/λ by Equivalently, using the column lengths defined in Definition 9.4, b(λ/λ ) = a(λ) j=1 . Observe that b(λ/λ ) is the weight of the tableau t(λ/λ ) having entries 0, 1, . . . λ j − 1 in column j, for 1 j a(λ). It is easily seen that this tableau is semistandard and has the minimum weight of any tableau in SSYT (λ/λ ).
Lemma 9.7. The specialization s λ/λ (1, q, . . . , q ) is unimodal and centrally symmetric about n 2 . Proof. Like any symmetric function, s λ/λ can be expressed as a linear combination of Schur functions labelled by partitions. The lemma therefore follows from the "moreover" part of Theorem 3.4.
Sufficiency. To illuminate the condition in Theorem 9.5, we prove a slightly stronger result.
Necessity. The following lemma implies that if s λ/λ is -irreducible then t(λ/λ ) has at most one bumpable box.

Two row, two column and hook equivalences
We say that a partition of hook shape (a + 1, Theorem 10.1. Let λ and µ be partitions that each, separately, have either precisely two rows, precisely two columns or are of proper hook shape. Let , m ∈ N be such that (λ) and m (µ). Then all plethystic equivalences λ ∼ m µ are listed in one of the cases in Table 1.
Proof. The proofs for each family in Table 1 are similar. We illustrate the method by finding all plethystic equivalences between a two-row non-hook partition λ = (a, b) and a proper hook µ = (c+1, 1 d ). By our assumption, 2 and m d+1. By Lemma 4.2 and Theorem 1.3 we may assume that c d. By Theorem 3.4(h), λ ∼ m µ if and only if there is an equality of multisets (C(λ) + + 1) ∪ H(µ) = (C(µ) + m + 1) ∪ H(λ). Equivalently, Comparing the greatest element of each side as in Proposition 4.7 shows that if equality holds then + a = m + c + 1, that is m = + a − c − 1. Substituting for m using this relation, and inserting a − b + 1 into each multisubset, we find that λ ∼ +a−c−1 µ if and only if  Table 1. All plethystic equivalences between partitions that, separately, have either precisely two rows, precisely two columns, or are of proper hook shape. In cases (j), (j ) and (k), (k ) the two hook partitions on the right are conjugates.
Equivalences between proper hook partitions: a 1, b 1 Equivalences between two-row non-hook partitions: a b 2 Equivalences between two-column non-hook partitions: a 2, c 2

Equivalences between a two-row non-hook and a proper hook partition
Equivalences between a two-column non-hook and a proper hook partition: a 2 (l) (2 a , 1 b ) ∼ a+b a+b ( We claim that this multiset equality implies a − c − d b. Indeed, if a − c − d > b then, on the left hand side, + 1 Therefore the multiplicity of + b − 1 is two, and comparing with the multiset on the right shows that = 1, contrary to our initial assumption. As a − c − d b, we may compare greatest elements of the above multisets to show that a + 1 = + b − 1, that is = a − b + 2. We substitute for using this relation Algebraic Combinatorics, Vol. 4 #1 (2021) and cancel the elements Since c + d + 1 c + 2 d + 2, for the multiset on the left to equal a union of two intervals each containing 1, we must have c = a − b + 2 and d = a − b. Then we have an equality of multisets if and only if c + d + 1 = b. We obtain the case with c d in (k), namely In the remaining case a − c − d 0, and so c + d a b. We cancel the elements + 1, . . . , + a from each side in (27)  We remark that the Haskell [25] software HookContent.hs available from the second author's website (2) was used to discover many of the equivalences appearing in Table 1. It has also been used to verify the more fiddly part of the authors' proof, by showing that every plethystic equivalence between two partitions of the types above, each of size at most 30, appears in our classification. Finally we observe that it follows from Proposition 3.6 and elementary number-theoretic arguments that the only plethystic equivalences in Table 1 involving distinct partitions that lift to isomorphisms of GL 2 (C)-representations are the infinite families
The result now follows from the two displayed equations.
11.2. Existence of exceptional equivalences. To prove Theorem 1.10(ii) we use the pyramid notation seen in Example 2.16, applied to the following partitions. The partition µ 60 is the lexicographically least partition in an exceptional equivalence when = 6. The other partitions were discovered by a computer search using the software already mentioned. The special case = 5 and m = 0 of the following proposition was seen in Example 2.16. Proof. It is clear from Definition 11.2 that λ m = µ m for any and m. Moreover, since the second part in each µ m is strictly smaller than a(µ m ), each partition µ •( +1) m has precisely parts. Since each partition λ m has precisely − 1 parts, it follows that λ m = µ For small this is a routine verification using the pyramid notation seen in Example 2.16. To illustrate the method we take = 8. It will be useful to say that a pyramid entry involves m if it of the form c + m for some c ∈ N. The difference sequences for λ m and η m are (2, 2, 1, 1, 1, 1 + m, 3, 1 −9 , 2, 1) (1, 1, 1, 2, 2, 1 + m, 1, 2, 1 −9 , 3), respectively. When = 8, the corresponding pyramids are Similarly one can check that ∆ (λ k ) = ∆ (η k ) for all k ∈ N 0 and all such that 5 18. This can be done programmatically using the Mathematica [30] notebook ExceptionalEquivalences.nb available from the second author's website. For the generic case when 18 we partition the pyramids P and Q for λ m and η m as shown in Figure 6. Using the calculation rule from § 2.7 one finds that the first 8 rows of the pyramid P for λ m are The final six rows of the pyramids (region C) are, with the constant factor + m removed, respectively. The corresponding difference multiset is { − 3}/{ − 2}. Hence the multisets of entries involving m in P and Q are the same.

Solitary partitions
Proposition 12.2. For each k ∈ N, the partition δ(k) is solitary.

H(µ) H δ(k) .
Let u be the number of boxes (i, j) ∈ [µ] such that h (i,j) (µ) = 2; we say that such boxes are 2-hooks. Since all the hook lengths in δ(k) are odd, the multiplicity of 2 in the right-hand side is u. By Lemma 4.4, µ has precisely k removable boxes. Since µ = δ(k), µ has at least one 2-hook, and so u 1. For any partition ν and n such that n (ν), we have 2 ∈ C(ν) + n + 1 if and only if n = (ν); in this case the multiplicity is 1. Therefore u 1 and we conclude that µ has a unique 2-hook. Moreover, m = (µ) and > k. Algebraic Combinatorics, Vol. 4 #1 (2021) To identify we use Proposition 4.7 to get a δ(k) + = a(µ) + m. (Or one may follow the proof of this proposition and instead compare greatest elements in (34).) Hence = a(µ) + m − k. Since µ has a unique 2-hook and precisely k removable boxes, it is obtained from δ(k) by inserting either (i) d new columns or (ii) d new rows of a fixed length c k. We consider these cases separately below. Observe that in either case the greatest hook length in either µ or δ(k) is (k + d − 1) + (k − 1) + 1 = 2k + d − 1, coming uniquely from the box (1, 1) of µ. Hence 2k + d − 1 has multiplicity 1 in the right-hand side of (34).