Sami H. Assaf, David E. Speyer Specht modules decompose as alternating sums of restrictions of Schur modules Proceedings of the American Mathematical Society DOI: 10.1090/proc/14815 Accepted Manuscript This is a preliminary PDF of the author-produced manuscript that has been peer-reviewed and accepted for publication. It has not been copyedited, proofread, or finalized by AMS Production staff. Once the accepted manuscript has been copyedited, proofread, and finalized by AMS Production staff, the article will be published in electronic form as a \Recently Published Article" before being placed in an issue. That electronically published article will become the Version of Record. This preliminary version is available to AMS members prior to publication of the Version of Record, and in limited cases it is also made accessible to everyone one year after the publication date of the Version of Record. The Version of Record is accessible to everyone five years after publication in an issue. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000{000 S 0002-9939(XX)0000-0 SPECHT MODULES DECOMPOSE AS ALTERNATING SUMS OF RESTRICTIONS OF SCHUR MODULES SAMI H. ASSAF AND DAVID E. SPEYER (Communicated by Benjamin Brubaker) Abstract. Schur modules give the irreducible polynomial representations of the general linear group GLt. Viewing the symmetric group St as a subgroup of GLt, we may restrict Schur modules to St and decompose the result into a direct sum of Specht modules, the irreducible representations of St. We give an equivariant M¨obiusinversion formula that we use to invert this expansion in the representation ring for St for t large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternate in sign by degree. In particular, this allows us to define a new basis of symmetric functions whose structure constants are stable Kronecker coefficients and which expand with signs alternating by degree into the Schur basis. 1. Overview of main results This paper concerns the relation between the representation theories of the gen- eral linear group GLt and the symmetric group St over C. To fix notation, for λ an integer partition, let `(λ) denote the length of λ (number of nonzero parts), and let SλS denote the size of λ (sum of the parts). Let Sλ denote the Schur functor, so t that the irreducible polynomial representations of GLt are Sλ(C ) where `(λ) ≤ t. For ν an integer partition, let Spν be the Specht module over C, so that the irreducible representations of St are Spν for SνS = t. t Since St ⊂ GLt, we can restrict the GLt representation Sλ(C ) to St and de- compose the result into Specht modules. For a partition ν = (ν1; ν2; : : : ; νr), and t t ≥ ν1 + SνS, we define ν( ) to be the partition (t − SνS; ν1; ν2; : : : ; νr) of t. Using this notation, we can write the aforementioned restriction as aν t (1.1) ResGLt ( t) ≅ ? Sp⊕ λ( ); St Sλ C ν(t) ν ν where aλ(t) are, by definition, the non-negative multiplicities that arise. In other words, in the representation ring Rep(St) of St, we have t ν (1.2) [Sλ(C )] = Q aλ(t)[Spν(t) ]: ν ν A classical result of Littlewood [5] states that aλ(t) is independent of t for t ν sufficiently large. Therefore we may define coefficients aλ by ν ν (1.3) aλ = lim aλ(t): t→∞ 2010 Mathematics Subject Classification. Primary 20C15; Secondary 20C30, 05E05, 05E10. SHA supported by NSF DMS-1763336; DES supported by NSF DMS-1600223. ©XXXX American Mathematical Society 1 This12 isJul a pre-publication 2019 00:08:38 version of thisEDT article, which may differ from the final published version. CopyrightAlgebra+NT+Comb+Logi restrictions may apply. Version 2 - Submitted to Proc. Amer. Math. Soc. 2 S. H. ASSAF AND D. E. SPEYER ν For λ, ν partitions with SλS ≤ SνS, Littlewood showed aλ = δλ,ν . In particular, we ν may regard aλ as entries of an infinite upper uni-triangular matrix with rows and columns indexed by partitions. It is natural to invert this matrix to define coeffi- ν cients bλ by ν ν −1 (1.4) [bλ] = [aλ] : ν ν While the aλ are non-negative, the bλ are, a priori, merely integers. Our main result is the following. SλS−SνS ν Theorem 1.1. With the above notation, we have (−1) bλ ≥ 0. ν As we explain in Section 1.2, the bλ have recently become of interest as part of a strategy for computing stable Kronecker coefficients, so this basic result concerning their signs seems of importance. ν 1.1. Plethystic formulas. We can give a precise formula for bλ using the language of plethysm. If ∶ GLm → GLn has character g and φ ∶ GLn → GLp has character f, then φ ○ ∶ GLm → GLp has character f[g], the plethysm of f and g. In terms α of symmetric polynomials, if g = ∑α gαx is the monomial expansion, then f[g] is f(y1; : : : ; yt), where the yi are defined by the identity α gα M (1 + yiq) = M (1 + x q) : α α In other words, if the gα are non-negative, then x occurs gα times in the multiset (y1; : : : ; yt). For more details on plethysm, see [8] and [9, (I.8)]. Littlewood [6] gave a formula for restriction from GLt to St as the following plethysm ν (1.5) aλ(t) = ⟨sλ; sν(t) [1 + h1 + h2 + ⋯]⟩; t where sλ(x1; : : : ; xt) = char(Sλ(C )) is the Schur polynomial corresponding to = the irreducible character for GLt, hn s(n) is the complete homogeneous symmetric polynomial, and the inner product for characters, corresponding to the Hall inner product on symmetric polynomials, is determined by ⟨sλ; sµ⟩ = δλ,µ. Define the Lyndon symmetric function Lm by 1 (1.6) L = Q µ(d)pm~d; m n d dSm The Lyndon symmetric function is the character of the GLt action on the degree m part of the free Lie algebra on t and is the Frobenius character of IndSm e2πi~m C Cm where Cm is the cyclic subgroup of Sm generated by the m-cycle (12⋯m). Using ν Lm, we can give an explicit formula for bλ as follows. Theorem 1.2. For λ and ν partitions, we have ν SνS−SλS (1.7) bλ = Q (−1) ⟨sµT ; sλT [L1 + L2 + L3 + ⋯]⟩; ν~µ vert. strip where λT denotes the transpose of the partition λ. We remark that L1 + L2 + L3 + ⋯ can be viewed as the GLt character of the free t Lie algebra on C . See [17] for a representation theoretic interpretation of this fact. t Our proofs involve an intermediate St-representation Mµ defined by t St (1.8) M = Ind Sp ⊠ 1t µ µ SSµS×St−SµS µ −S S This12 isJul a pre-publication 2019 00:08:38 version of thisEDT article, which may differ from the final published version. CopyrightAlgebra+NT+Comb+Logi restrictions may apply. Version 2 - Submitted to Proc. Amer. Math. Soc. SPECHTS ARE ALTERNATING IN SCHURS 3 where 1k is the trivial representation of Sk. t Since Mµ is an St representation, it is a positive combination of the Specht t t modules Spνt . We will show that, in turn, Sλ(C ) is positive in the Mµ. We derive t our result (1.7) by composing a formula for Spνt in terms of the Mµ and a formula t t for Mµ in terms of Sλ(C ). The following theorem gives plethystic formulas for transitioning between each of these bases. Theorem 1.3. In the representation ring Rep(St), we have: t (1.9) [Mµ] = Q⟨sν ; sµ[1 + h1]⟩[Spνt ] = Q [Spν(t) ]: ν µ~ν horiz. strip t SνS−SµS t (1.10) [Spν(t) ] = Q⟨sµT ; sνT [−1 + h1]⟩[Mµ] = Q (−1) [Mµ]: µ ν~µ vert. strip t t (1.11)[Sλ(C )] = Q⟨sλ; sµ[h1 + h2 + h3 + ⋯]⟩[Mµ]: µ t SµS−SλS t (1.12) [Mµ] = Q(−1) ⟨sµT ; sλT [L1 + L2 + L3 + ⋯]⟩[Sλ(C )]: λ t The representation Mµ arises naturally in studying representations of the cat- egory of finite sets. A representation of the category of finite sets consists of a sequence of vector spaces V0, V1, V2, . and, for each map φ ∶ {1; 2; : : : ; t} Ð→ {1; 2; : : : ; u} of finite sets, a map φ∗ ∶ Vt → Vu obeying the obvious functoriality. In particular, each Vt is a representation of the symmetric group St. The category of such representations is an abelian category in an obvious manner. The simple objects in this category are explicitly described by Rains [15] and are implicitly described in the work of Putcha [14]; see also Wiltshire-Gordon [22]. These simple k objects Wµ are indexed by partitions and, except when µ is of the form 1 , we t have (Wµ)t ≅ Mµ as an St-representation. Wiltshire-Gordon also showed that the t Sλ(C ) are projective objects in this category. Thus, the problem of expanding t t Sλ(C ) positively in Mµ is the problem of finding the Jordan-Holder constituents t of these projectives, and the problem of writing Mµ as an alternating combina- t tion of the Sλ(C ) is a combinatorial shadow of the problem of finding projective resolutions of these simples. Our proofs generally utilize the representation theoretic interpretations of for- mulas over pure combinatorial manipulation with the one exception of proving Eq. (1.12); see the discussion preceding Theorem 6.3. 1.2. Stable Kronecker coefficients. The authors' original motivation for study- ing this problem came from a desire to understand tensor product multiplicities.