Obstructions to combinatorial formulas for plethysm

Motivated by questions of Mulmuley and Stanley we investigate quasi-polynomials arising in formulas for plethysm. We demonstrate, on the examples of $S^3(S^k)$ and $S^k(S^3)$, that these need not be counting functions of inhomogeneous polytopes of dimension equal to the degree of the quasi-polynomial. It follows that these functions are not, in general, counting functions of lattice points in any scaled convex bodies, even when restricted to single rays. Our results also apply to special rectangular Kronecker coefficients.

In this paper we study the more complicated plethysm coefficients m d,k λ defined by where the sum is over partitions λ of dk.
The computation of the multiplicity functions in plethysms can be seen as an operation on Schur polynomials [Mac98]. Viewed like this, it is surprising that when the plethysm is written in terms of other Schur polynomials, the coefficients are always nonnegative. Of course this must be true, because the coefficients are multiplicities of irreducible representations, but it would be desirable to have a combinatorial explanation for nonnegativity. In [Sta00, Problem 9] Richard Stanley asked for a positive combinatorial method to compute plethysm coefficients. A connection between plethysm and lattice point counting was shown at least in [KM16,Col17,CDW12]. These connections are not direct in the sense that plethysm coefficients are not seen to equal counts, but always involve some opaque arithmetics.
The functions f (s) = m d,sk sλ and g(s) = m sd,k sλ share many properties with Ehrhart functions of rational polytopes: • Both f and g are nonnegative and f (0) = g(0) = 1. • For regular λ, the leading term of f is proportional to the Ehrhart function of a rational polytope corresponding to the Littlewood-Richardson rule [KM16, Section 4].
Remark 1. The scaling factor between the leading coefficient of f and the Ehrhart function in the third bullet is the Plancherel measure on Young diagrams. It is conjectured that the same is true for any λ, however we are not aware of a proof.
Example 2. The multiplicity of λ = s · (2k − 1, 1) inside S 2 (S sk ) equals 0 when s is odd and 1 when s even. In particular, it is equal to the number of integral points in the (zero-dimensional) polytope {s/2}. The multiplicity of s · (2k, k) inside S 3 (S sk ) is equal to the number of integral points in the (one-dimensional) polytope s · [1/3, 1/2] or s · [1/2, 2/3]. Ketan Mulmuley posed several conjectures concerning the behavior of plethysm and Kronecker coefficients in relation to geometric complexity theory [MS01,Mul09]. The question about m sd,k sλ comes directly from [Mul09, Hypothesis 1.6.4]. We give a negative answer for both functions in Remark 6. Our main goal, however, is a generalized version of these questions for which we need some additional terminology. Following [Mul11, Section 5.1] we define a shifted or inhomogeneous rational polytope as a system of inequalities where b, c are arbitrary rational vectors and A is a rational matrix. Splitting the right hand side as b + c is motivated by the definition of the dilations of P as An asymptotic Ehrhart quasi-polynomial is a counting function of the form The dimension of a shifted rational polytope P is by definition the dimension of P (A, sb, c) for large s (for small s the polytope can be empty). Contrary to the case of Ehrhart quasi-polynomials, an asymptotic Ehrhart quasi-polynomial does not need to be a quasipolynomial, although it is for large arguments. Moreover, a quasi-polynomial may be an asymptotic Ehrhart quasi-polynomial but not an Ehrhart quasi-polynomial. Asymptotic Ehrhart quasi-polynomials do not have to satisfy Ehrhart reciprocity. Further, the dimension of a shifted rational polytope can be strictly greater than the degree of the associated asymptotic Ehrhart quasi-polynomial. See [Sta82] and [Sta96, Chapter I] for structure theory of asymptotic Ehrhart quasi-polynomials and the relation to linear diophantine equations.
In [Mul11, Hypothesis 5.3] it is conjectured that the multiplicity of s · π in S s·λ (S µ ) is an asymptotic Ehrhart quasi-polynomial with additional complexity-theoretic properties. The asymptotic part of the conjecture is motivated by the failure of Ehrhart type explanations already for Kronecker coefficients [BOR09,KW09] which are often considered less complicated than plethysm coefficients. In the present paper we make progress towards a negative answer of the following simplified version.
Our main result, Theorem 5, implies a negative answer to both cases in Question 3 and strong restrictions on a positive solution to Question 4. We show that m sd,k sλ need not be an asymptotic Ehrhart quasi-polynomial of a polytope of dimension equal to the degree of its growth. At the moment we are not able either to exclude or confirm that m sd,k sλ is an asymptotic Ehrhart quasi-polynomial for an inhomogeneous polytope of dimension strictly larger than its degree.
In previous work the authors gave a formula for plethysm coefficients, which is a sum of Ehrhart functions of various polytopes with (positive and negative) coefficients [KM16]. This allows to gather experimental data on the questions for many rays. A-posteriori, the specific plethysm in Theorem 5 can also be confirmed using well-known formulas for S 3 (S k ). General formulas for S k (S 3 ) are unknown, though. Our methods are inspired by [KW09,BOR09]. Theorem 5. The multiplicity functions of S s·(7,5,0) in S 3 (S 4s ) and S 4s (S 3 ) equal where r(s) has period 6 and takes the values 3, −1, 1, 0, 2, −2 on respectively the integers 0, . . . , 5. There do not exist rational vectors a, b, c ∈ Q n (of arbitrary length n) such that φ equals the counting function of a one-dimensional inhomogeneous polytope P (a, b, c). In particular, φ is not an Ehrhart function of any rational polyhedron.
Remark 6. Before giving a general proof, it is easy to see that φ cannot be an Ehrhart function: it violates Ehrhart-Macdonald reciprocity [Ehr67,Mac71]. The value f (−s) of an Ehrhart quasi-polynomial f of a rational polytope P at a negative integer argument counts (up to a global sign) the number of interior lattice points in s · P . In particular, Interestingly, the jumps in values that make Ehrhart reciprocity fail are also the crucial ingredient for our proof of non-representability by a one-dimensional inhomogeneous polytope.
Proof. The equality of the multiplicities of the given ray in both plethysms is a consequence of Hermite reciprocity [Her54], [FH91, Exercise 6.18]. There are now various ways to determine the formula from the interpretation as a multiplicity in S 3 (S 4s ). One is to simply evaluate the explicit formula from [KM16] along a ray. Another way is to observe that the function must exhibit linear growth and that its period is at most six [CGR84,MM15]. With this information and some values the function can be interpolated. Let P be an inhomogeneous line segment as in the statement. Then the s-th dilation of P can be written as Asymptotically, P (s) becomes an interval of length s 3 . This means that there exists an s 0 such that for all s > s 0 for some b, c, c ∈ Q. Making s 0 even larger it can be assumed that s 0 b are integers and that s 0 is divisible by 6 (the period of φ). Since #Q(s) and #P (s) have the same linear term and constants that are s 0 -periodic (by the divisibility assumptions) they agree for all s. Indeed, changing b by an integer also only shifts Q by an integral value. As φ(5) + 2 = φ(6) there must be at least two integers x such that 5(b + 1 3 ) + c < x 6(b + 1 3 ) + c . Hence, b + 1 3 > 1. In particular, for any s there is always at least one integer x satisfying s(b + 1 3 ) + c < x (s + 1)(b + 1 3 ) + c (an interval of length > 1 must contain an integer). On the other hand, by (2) there is at most one integer x satisfying sb + c < x (s + 1)b + c (an interval of length < 1 may contain at most one integer). The above two intervals are the difference between P (s) and P (s + 1). It follows that φ(s) is nondecreasing. This contradicts φ(0) = 1 > 0 = φ(1).
Remark 7. We have confirmed that all examples with two rows and fewer boxes are in fact Ehrhart functions. In this sense, our counter-example is minimal. Remark 9. While this is not visible from the representation in [KM16], a general theorem of Meinrenken and Sjamaar [MS99] implies that the chambers of plethysm quasipolynomials are polyhedral cones (see [KM16,Remark 3.12], [PV15]). One may thus ask the stronger question if they are Ehrhart quasi-polynomials in general. Of course, Theorem 5 gives a negative answer to this much stronger property too.
Lattice point counting in polytopes and Ehrhart functions are arguably the most natural combinatorial explanations one may hope for here, but they are not the only ones. Our theorem also excludes other possibilities. Proof. One-dimensional convex sets are polyhedra and the counting function in Theorem 5 would need to specialize to a positive combination of Ehrhart functions.
The plethysm counting function in Theorem 5 cannot be written as a positive combination of Ehrhart quasi-polynomials as these also have to satisfy Ehrhart reciprocity. However, it can be written as a sum of an asymptotic Ehrhart quasi-polynomial and an honest Ehrhart quasi-polynomial as in the following proposition (which is very easy to confirm).
Proposition 11. Fix a small rational ε > 0. Let P (s) = [ε, s · 1 3 + ε] and Q(s) = s · { 1 2 }. Then φ(s) = #(P (s) ∩ Z) + #(Q(s) ∩ Z). test if the quasi-polynomial is an (asymptotic) Ehrhart function? Assume it is not an Ehrhart function globally, but is an Ehrhart function on each individual ray. Is there a finite number of rays such that finding polytopes for each of the finitely many rays proves the Ehrhart nature for all rays? Is it some multi-dimensional generalization of Apéry sets known from numerical semigroups?
Recently, there has been a lot of interest in Kronecker coefficients [Man16,BOR09,SS16,BV15]. These are different from plethysm, but also hard to compute and important in geometric complexity theory. It is known that on rays they are given by quasi-polynomials [Man15, Theorem 1] that do not have to be Ehrhart functions [KW09]. However, the problem is open for specific rays that are important in GCT. Emmanuel Briand pointed us towards following observation. Two row partitions are useful because Hermite reciprocity applies to them. Further, by the work of Vallejo [Val14], cf. [Mac98], the plethysm coefficients in Theorem 5 are equal to special Kronecker coefficients: φ(s) = K(3 4s , 3 3s , (7s, 5s)) = K((4s, 4s, 4s), (4s, 4s, 4s), (7s, 5s)), where K depends on three partitions and is the Kronecker coefficient. Indeed, let p n (a, b) denote the number of Young diagrams with n boxes contained inside an a × b rectangle. Both the multiplicity of λ with two rows in the plethysm S a (S b ) and the Kronecker coefficient K(a b , a b , λ) can be expressed as the difference p λ 2 (a, b) − p λ 2 −1 (a, b).
For example by transposition, this immediately implies Hermite reciprocity.
In a different direction, the following interesting question was recently brought to our attention by Michèle Vergne.
One can prove that the answer is positive when f or g are of degree one, thus to find a counterexample one needs to consider partitions with more than one row.