Decomposition matrices, positive-characteristic character formulas, and the extension-of-scalars map for non-split reductive group schemes (open questions from Serre 1968)

In a 1968 paper on the Grothendieck group of split reductive group schemes, Serre proved that for such a group scheme $G$ over a principal ideal domain $A$ the extension-of-scalars map $i:R_A(G)\to R_K(G)$ and the decomposition homomorphisms $d_v:R_K(G)\to R_{k_v}(G)$ are isomorphisms, and left three questions open: (a) determine explicitly the decomposition matrix of $d_v$ in mixed characteristic ($\mathrm{char}\,K=0$, $\mathrm{char}\,k_v=p>0$); (b) give, for a split reductive group over a field of positive characteristic, an explicit analogue of Weyl's character formula for the characters of simple modules; (c) for reductive group schemes that are not necessarily split, decide whether $i$ is an isomorphism (the $d_v$ being then injective but not generally surjective). Questions (a) and (b) coincide with the central open problem of the modular representation theory of reductive algebraic groups in defining characteristic: computing the composition multiplicities of Weyl modules, equivalently the characters of simple modules. The later literature (Steinberg's tensor product theorem; Lusztig's conjecture; the large-$p$ theorems of Andersen–Jantzen–Soergel and Fiebig; Williamson's counterexamples in small characteristic; the $p$-canonical basis character formulas of Achar–Makisumi–Riche–Williamson and Riche–Williamson, which settle the large-$p$ regime $p\ge 2h-2$, with $h$ the Coxeter number) leaves the general case — in particular the regime of characteristic small relative to the rank — open, and no resolution of (c) was found.

Background

To any exact category one associates its Grothendieck group: the free abelian group on isomorphism classes of objects, modulo one relation $[B]=[A]+[C]$ for every short exact sequence $0\to A\to B\to C\to 0$. For an affine group scheme $G$ over a commutative ring $A$ — a group object represented by a Hopf algebra $A[G]$ of functions — one takes the category of comodules over $A[G]$ whose underlying $A$-modules are finite projective; its Grothendieck group is written $R_A(G)$. When $A$ is a field this is the classical Grothendieck group of finite-dimensional rational representations of the algebraic group $G$, and its standard basis is the set of classes of simple (irreducible) representations.

The source paper (Serre 1968) studies the following situation, which is also the framework of the problems below. Let $A$ be a principal ideal domain, $K$ its field of fractions, $V$ the set of its nonzero prime ideals $v$, and $k_v=A/v$ the residue field; the localization $A_v$ is a discrete valuation ring, i.e., a local principal ideal domain. Extension of scalars defines a homomorphism $i:R_A(G)\to R_K(G)$. For each $v$, one chooses a $G$-stable lattice (a $G$-stable finite projective $A$-submodule spanning the representation over $K$) and reduces it modulo $v$; passage to Grothendieck classes defines the decomposition homomorphism $d_v:R_K(G)\to R_{k_v}(G)$, independently of the choice of lattice. This is Brauer's classical decomposition theory for finite groups, extended in the paper to comodules over arbitrary flat coalgebras. The motivating question, attributed to Grothendieck, was whether $i:R_{\mathbb Z}(\mathrm{GL}_n)\to R_{\mathbb Q}(\mathrm{GL}_n)$ is an isomorphism, which matters for the universal role of $R_{\mathbb Z}(\mathrm{GL}_n)$ in the theory of $\lambda$-operations (the operations on Grothendieck groups induced by exterior powers) on $K$-theory.

A reductive group scheme over $A$ is a smooth affine group scheme whose fibers over $\mathrm{Spec}\,A$ are connected reductive algebraic groups, such as $\mathrm{GL}_n$, $\mathrm{SL}_n$ or $\mathrm{Sp}_{2n}$; it is called split (French: déployé) if it contains a split maximal torus over $A$ together with root subgroups defined over $A$. Let $M$ be the character group of the split maximal torus $T$ (a free abelian group of finite rank) and $W$ the Weyl group of $G$ (the finite reflection group generated by the reflections in the roots). Serre's Theorem 5 states that for $G$ split reductive over the principal ideal domain $A$, both $i$ and all $d_v$ are isomorphisms; in particular $R_A(G)$ is torsion-free. Over a field $k$, restricting a representation of a split $G$ to $T$ and taking weights defines the character map $\mathrm{ch}_G:R_k(G)\to\mathbb Z[M]$, and the paper's Theorem 4 identifies $R_k(G)$ with the invariants $\mathbb Z[M]^W$.

The simple modules of a split reductive group over $k$ are indexed by the dominant weights (weights $\lambda\in M$ nonnegative on each coroot); write $L(\lambda)$ for the simple module of dominant highest weight $\lambda$. In characteristic $0$, $\mathrm{ch}_G(L(\lambda))$ is given by Weyl's character formula, the classical explicit alternating sum over the Weyl group (a scheme-theoretic proof was later given by Demazure); in positive characteristic no general formula of comparable explicitness is known (the known partial results are described at the end of this paragraph). Serre's decomposition matrix links the two characteristics. Suppose $\mathrm{char}\,K=0$ and $\mathrm{char}\,k_v=p>0$. The matrix of $d_v$ in the bases of simple modules is unitriangular (triangular, with all diagonal entries equal to $1$) for the natural partial order on weights; reducing a $G$-stable lattice in the characteristic-zero simple module $L(\lambda)_K$ modulo $v$ produces the Weyl module $\Delta(\lambda)$ over $k_v$ (also called a standard module; generally not simple), and the entries of the matrix are the composition multiplicities $[\Delta(\lambda):L(\mu)]$, the defining-characteristic decomposition numbers. Since $\mathrm{ch}_G(\Delta(\lambda))$ is still given by Weyl's character formula, and the classes of the $L(\mu)$ form a basis of $R_{k_v}(G)$, knowing all decomposition numbers is equivalent to knowing all characters $\mathrm{ch}_G(L(\mu))$ in positive characteristic, and conversely. When $\mathrm{char}\,K=\mathrm{char}\,k_v$ the decomposition matrix is the identity and there is nothing to compute.

What is known about this core is collected in Jantzen's monograph: in rank one the decomposition numbers are completely known; Steinberg's tensor product theorem factorizes $L(\lambda)$ according to the $p$-adic digits of $\lambda$; the linkage principle restricts which composition factors can occur; the Jantzen sum formula gives a nontrivial alternating constraint on the multiplicities. Lusztig's conjecture (1980) predicted a formula for $\mathrm{ch}(L(\lambda))$, in a natural range of low weights, in terms of Kazhdan–Lusztig polynomials — combinatorial polynomials attached to the Iwahori–Hecke algebra (a one-parameter deformation of the group algebra) of the affine Weyl group (the semidirect product of the space of translations by the coroot lattice with the Weyl group $W$) — when $p$ is at least the Coxeter number (the order of a Coxeter element of $W$, equivalently one plus the largest height of a root). Andersen–Jantzen–Soergel (1994) proved the conjecture for $p$ sufficiently large, with an ineffective bound; Fiebig (2011) gave the first effective bounds, still enormous; Williamson (2017) constructed many counterexamples for $\mathrm{SL}_n$ when $p$ is small relative to the rank, so the conjectured formula is false in general. The current best general framework expresses characters through the $p$-canonical basis of the affine Hecke category (the basis built from parity sheaves on affine flag varieties rather than from intersection cohomology). Riche–Williamson introduced this framework and proved the translation-functor statement it requires for $\mathrm{GL}_n$; Achar–Makisumi–Riche–Williamson (2019) then proved the character formula for indecomposable tilting modules (the indecomposable modules admitting both a filtration by Weyl modules and a filtration by their duals) unconditionally for $p>h$, and stated that, using results of Andersen recorded there, one obtains a character formula for simple modules for $p\ge 2h-2$; Riche–Williamson (2021) proved directly a character formula for the simple modules in the principal block (the block containing the trivial representation), valid for $p>2h-1$, presented as a replacement for Lusztig's conjecture. These formulas settle the regime of large characteristic relative to the Coxeter number, but in the regime of characteristic small relative to the rank — where Williamson's counterexamples live — no general formula or algorithm is known.

The third question concerns the map $i$ without the splitness hypothesis. For $G$ reductive but not necessarily split over $A$, the $d_v$ are injective but not generally surjective, and the paper states that nothing is known about $i$ in this case. A warning from the same paper: for split $G$ over a Dedekind domain $A$ one has $R_A(G)\cong K_0(A)\otimes R_K(G)$, where $K_0(A)$ is the Grothendieck group of finite projective $A$-modules, isomorphic to $\mathbb Z\oplus Cl(A)$ with $Cl(A)$ the ideal class group (the group measuring the failure of unique factorization of fractional ideals); hence $i$ already fails injectivity when $Cl(A)\ne 0$, and the principal ideal domain hypothesis is essential. The non-split case over a principal ideal domain is the natural open frontier. Adjacent literature develops Tannakian duality over rings (Wedhorn over valuation rings; Nguyen Dai Duong and Phùng Hô Hai over Dedekind rings) — reconstructing a group scheme from its category of integral representations — and an isomorphism theorem for the Grothendieck ring of a Hopf algebra order (an order being a finite module over the base ring spanning the algebra) in the group algebra of a finite abelian group (Jensen–Larson), but none of these computes the map $i:R_A(G)\to R_K(G)$ for non-split reductive group schemes.

Problem Statement

Work in the framework of the source paper: $A$ is a principal ideal domain with fraction field $K$, $V$ its set of nonzero prime ideals, $k_v$ the residue field at $v\in V$, and $G$ a reductive group scheme over $A$ (representations understood on finite projective $A$-modules, equivalently comodules over the coordinate Hopf algebra $A[G]$). It is known that if $G$ is split, then $i:R_A(G)\to R_K(G)$ and all $d_v:R_K(G)\to R_{k_v}(G)$ are isomorphisms. The open problems:

(a) Decomposition matrix in mixed characteristic. For $G$ split reductive over $A$ with $\mathrm{char}\,K=0$ and $\mathrm{char}\,k_v=p\ne 0$: determine explicitly the matrix of $d_v$ with respect to the bases of simple modules on both sides; equivalently, compute all defining-characteristic decomposition numbers $[\Delta(\lambda):L(\mu)]$ of the Weyl modules $\Delta(\lambda)$ over $k_v$, for arbitrary split reductive $G$, arbitrary $p>0$, and arbitrary dominant weights $\lambda,\mu$. (When $\mathrm{char}\,K=\mathrm{char}\,k_v$ the matrix is the identity, so the content is entirely in the case of nonzero residue characteristic.)

(b) Positive-characteristic character formula. For a split reductive group over an arbitrary field $k$ of characteristic $p\ne 0$: give an explicit formula, analogous to Weyl's character formula, for the character $\mathrm{ch}_G(L(\lambda))$ of the simple module of dominant highest weight $\lambda$. By unitriangularity of $d_v$ and Weyl's formula for $\mathrm{ch}_G(\Delta(\lambda))$, parts (a) and (b) are equivalent in content whenever the characteristic-$p$ field arises as a residue field; a solution of either yields the other.

(c) The map $i$ in the non-split case. For $G$ a reductive group scheme over $A$ that is not necessarily split — in which case the $d_v$ are injective but not surjective in general — determine whether the extension-of-scalars homomorphism $i:R_A(G)\to R_K(G)$ is an isomorphism — that is, whether every class of a rational representation of $G_K$ lies in the image of $i$ (surjectivity), and whether $i$ has nonzero kernel, so that a nonzero element of $R_A(G)$ vanishes after extension of scalars (injectivity).

The common objective of (a)–(c) is to make the source paper's qualitative isomorphism theorems effective and to extend them beyond the split case: explicit computation of the invariants $i$ and $d_v$ describe for split groups, and structural extension of the isomorphism $i$ to non-split reductive group schemes. Each part also admits independent partial progress.

References

1. Jean-Pierre Serre, Groupe de Grothendieck des schémas en groupes réductifs déployés, Publications Mathématiques de l'IHÉS 34 (1968), 37–52. DOI: 10.1007/BF02684589, https://doi.org/10.1007/BF02684589 (open-access full text: https://www.numdam.org/item/PMIHES_1968__34__37_0/)
2. Michel Demazure, Sur la formule des caractères de H. Weyl, Inventiones Mathematicae 9 (1970), 249–252. DOI: 10.1007/BF01404328, https://doi.org/10.1007/BF01404328
3. Jens Carsten Jantzen, Representations of Algebraic Groups, 2nd edition, Mathematical Surveys and Monographs 107, American Mathematical Society, Providence, RI, 2003. DOI: 10.1090/surv/107, https://doi.org/10.1090/surv/107
4. George Lusztig, Some problems in the representation theory of finite Chevalley groups, in: The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proceedings of Symposia in Pure Mathematics 37, American Mathematical Society, Providence, RI, 1980, pp. 313–317. DOI: 10.1090/pspum/037/604598, https://doi.org/10.1090/pspum/037/604598 (zbMATH: 0453.20005)
5. Henning Haahr Andersen, Jens Carsten Jantzen, Wolfgang Soergel, Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: independence of p, Astérisque 220, Société Mathématique de France, Paris, 1994, 323 pp. NumDam: AST_1994__220__1_0, https://www.numdam.org/item/AST_1994__220__1_0/ (zbMATH: 0802.17009)
6. Peter Fiebig, Sheaves on affine Schubert varieties, modular representations, and Lusztig's conjecture, Journal of the American Mathematical Society 24 (2011), 133–181. DOI: 10.1090/S0894-0347-2010-00679-0, https://doi.org/10.1090/S0894-0347-2010-00679-0 (preprint arXiv:0711.0871, https://arxiv.org/abs/0711.0871)
7. Geordie Williamson, Schubert calculus and torsion explosion, Journal of the American Mathematical Society 30 (2017), 1023–1046. DOI: 10.1090/jams/868, https://doi.org/10.1090/jams/868 (preprint arXiv:1309.5055, https://arxiv.org/abs/1309.5055)
8. Simon Riche, Geordie Williamson, Tilting modules and the p-canonical basis, Astérisque 397, Société Mathématique de France, Paris, 2018. DOI: 10.24033/ast.1041, https://doi.org/10.24033/ast.1041 (preprint arXiv:1512.08296, https://arxiv.org/abs/1512.08296)
9. Anna-Lise Jensen, Richard G. Larson, An isomorphism for the Grothendieck ring of a Hopf algebra order, Proceedings of the American Mathematical Society 97 (1986), no. 2, 197–200. DOI: 10.1090/S0002-9939-1986-0835864-3, https://doi.org/10.1090/S0002-9939-1986-0835864-3
10. Torsten Wedhorn, On Tannakian duality over valuation rings, Journal of Algebra 282 (2004), 575–609. DOI: 10.1016/j.jalgebra.2004.07.024, https://doi.org/10.1016/j.jalgebra.2004.07.024
11. Nguyen Dai Duong, Phùng Hô Hai, Tannakian duality over Dedekind rings and applications, Mathematische Zeitschrift 288 (2018), 1103–1142. DOI: 10.1007/s00209-017-1928-6, https://doi.org/10.1007/s00209-017-1928-6
12. Phùng Hô Hai, João Pedro dos Santos, Finite torsors on projective schemes defined over a discrete valuation ring, Algebraic Geometry 10 (2023), no. 1, 1–40. DOI: 10.14231/ag-2023-001, https://doi.org/10.14231/ag-2023-001 (preprint arXiv:1904.10659, https://arxiv.org/abs/1904.10659)
13. Pramod N. Achar, Shotaro Makisumi, Simon Riche, Geordie Williamson, Koszul duality for Kac–Moody groups and characters of tilting modules, Journal of the American Mathematical Society 32 (2019), no. 1, 261–310. DOI: 10.1090/jams/905, https://doi.org/10.1090/jams/905 (preprint arXiv:1706.00183, https://arxiv.org/abs/1706.00183)
14. Simon Riche, Geordie Williamson, A simple character formula, Annales Henri Lebesgue 4 (2021), 503–535. DOI: 10.5802/ahl.79, https://doi.org/10.5802/ahl.79 (preprint arXiv:1904.08085, https://arxiv.org/abs/1904.08085)