Choosing the Broué abelian-defect equivalence to satisfy pointwise character-degree inequalities

For a block $B$ of a finite group algebra over a $p$-modular system with abelian defect group $P$, Broué's Abelian Defect Group Conjecture predicts a derived equivalence, and hence a perfect isometry, between $B$ and its Brauer correspondent $C$ on $N_G(P)$. Linckelmann (2025), after proving Giannelli's degree-refined McKay conjecture (Conjecture B) for cyclic and Klein four defect blocks, asked whether the predicted equivalence can always be chosen so that its induced character bijection $f$ satisfies the pointwise inequality $f(\chi)(1)\le\chi(1)$ for every $\chi$, or possibly even the stronger source-algebra inequality $f(\chi)(1)\le\chi(i)$. The question is posed for all blocks with abelian defect groups. A positive answer would entail Broué's conjecture itself together with Giannelli's Conjecture B in the abelian-defect case; the question is open, with positive answers known only for nilpotent, inertial, cyclic-defect, Klein-four-defect, and closely related restricted families.

Background

Fix a prime $p$ and a $p$-modular system $(\mathcal{O},K,k)$: $\mathcal{O}$ is a complete discrete valuation ring with field of fractions $K$ of characteristic $0$ and algebraically closed residue field $k$ of characteristic $p$. For a finite group $G$, the group algebra $\mathcal{O}G$ decomposes as a direct product of indecomposable two-sided ideals called blocks. Each block $B$ of $\mathcal{O}G$ determines, up to conjugacy, a $p$-subgroup $P$ of $G$ called its defect group, which measures the $p$-part of the representations belonging to $B$; the block of $\mathcal{O}N_G(P)$ paired with $B$ by Brauer's main theorems is the Brauer correspondent $C$. Writing $\mathrm{Irr}(B)$ for the set of irreducible ordinary characters belonging to $B$, the height of $\chi\in\mathrm{Irr}(B)$ is the integer $h$ with $\chi(1)_p=(|G|_p/|P|)\cdot p^h$; height-zero characters are those with $\chi(1)_p=|G|_p/|P|$. The McKay conjecture (recently proved in general by Cabanes and Späth) predicts that the set $\mathrm{Irr}^{p'}(G)$ of irreducible characters of $G$ of degree prime to $p$ has the same cardinality as its normalizer analogue $\mathrm{Irr}^{p'}(N_G(P))$ for a Sylow $p$-subgroup $P$, and its block-wise refinement, the Alperin–McKay conjecture, predicts a bijection $\mathrm{Irr}^0(B)\cong\mathrm{Irr}^0(C)$ between the height-zero characters of a block and of its Brauer correspondent. By the relevant direction of Brauer's height-zero conjecture, established by Kessar and Malle, a block with abelian defect group has all characters of height zero, so for abelian defect $\mathrm{Irr}^0(B)=\mathrm{Irr}(B)$.

Giannelli (2025) proposed strengthening the McKay correspondence by a numerical condition: Conjecture A predicts a bijection $\varepsilon:\mathrm{Irr}^{p'}(G)\to\mathrm{Irr}^{p'}(N_G(P))$ with $\varepsilon(\chi)(1)\le\chi(1)$ for all $\chi$, and Conjecture B the analogous bijection $\varepsilon:\mathrm{Irr}^0(B)\to\mathrm{Irr}^0(b)$ between a block and its Brauer correspondent. Conjecture A is proved for symmetric groups, and was reduced to finite simple groups and proved for $p=2$ by Hung, Martínez and Navarro; Conjecture B is known for $p$-solvable groups (finite groups built from $p$-groups and $p'$-groups by extensions; Martínez–Rossi, extending divisibility results of Geck, Rizo and Turull) and, by Linckelmann (2025, reference 1), for blocks with cyclic or Klein four defect group.

A perfect isometry (Broué, 1990) is a $\mathbb{Z}$-linear isometry $\Phi:\mathrm{ZIrr}(B)\to\mathrm{ZIrr}(C)$ induced by a signed bijection $f$ of irreducible characters that satisfies arithmetic conditions tying character values at $p$-elements and $p'$-elements of the two groups; it is the character-theoretic shadow of a derived equivalence between $B$ and $C$, that is, an equivalence of bounded derived categories of module categories, expected to come from a two-term complex of bimodules. Broué's Abelian Defect Group Conjecture predicts that whenever the defect group $P$ is abelian, $B$ and $C$ are derived equivalent, hence perfectly isometric; it is open in general and has been verified case by case since Dade's theory of blocks with cyclic defect groups (1966) and Brauer's analysis of Klein four defect groups (1971). A source idempotent $i$ of $B$ is a primitive idempotent in the $P$-fixed subalgebra $B^P$ whose Brauer homomorphism image is nonzero; the source algebra $iBi$ is a small interior $P$-algebra (an $\mathcal{O}$-algebra endowed with a group homomorphism from $P$ into its group of units) Morita equivalent to $B$, that is, with a module category equivalent to that of $B$, and each $\chi\in\mathrm{Irr}(B)$ corresponds to a simple $K\otimes_{\mathcal{O}}iBi$-module $iX$ of dimension $\chi(i)=\dim_K(iX)\le\chi(1)$. The inertial quotient of $B$ is the group $E=N_G(P,e)/PC_G(P)$ attached to a maximal Brauer pair $(P,e)$; it is a $p'$-subgroup of $\mathrm{Aut}(P)$.

Linckelmann (2025) confirmed Giannelli's Conjecture B, realized by perfect isometries, for blocks with cyclic defect group, for $p=2$ with Klein four defect group, and more generally for nilpotent blocks (blocks with trivial inertial quotient; via Puig's structure theory), for inertial blocks (blocks Morita equivalent to their Brauer correspondent via a bimodule with endopermutation source; a source algebra of such a block has the form $S\otimes_{\mathcal{O}}\mathcal{O}_\alpha(P\rtimes E)$, where $\mathcal{O}_\alpha(P\rtimes E)$ is the group algebra of the semidirect product possibly twisted by a 2-cocycle $\alpha$), and for blocks with abelian defect group and cyclic inertial quotient $E$ acting freely on $P\setminus\{1\}$ with either $p$ odd or $|E|=|P|-1$, assuming the counting equality $|\mathrm{Irr}(B)|=|\mathrm{Irr}(P\rtimes E)|$ (known for cyclic defect by Dade and for Klein four by Brauer). His Remark 4.3 then poses the question of whether the equivalence predicted by Broué's conjecture can, in general, be chosen to carry these degree inequalities — the natural fusion of the two conjectural pictures.

Problem Statement

Let $p$ be a prime and $(\mathcal{O},K,k)$ a $p$-modular system with $k$ algebraically closed. Let $G$ be a finite group, $B$ a block of $\mathcal{O}G$ with an abelian defect group $P$ (the case $P=1$ being trivial, one may take $P$ nontrivial), $C$ the Brauer correspondent of $B$ in $\mathcal{O}N_G(P)$, and assume $K$ is a splitting field for $B$ and $C$. Broué's Abelian Defect Group Conjecture predicts a derived equivalence between $B$ and $C$, inducing a perfect isometry $\Phi:\mathrm{ZIrr}(B)\to\mathrm{ZIrr}(C)$ with bijection $f:\mathrm{Irr}(B)\to\mathrm{Irr}(C)$. Determine whether such an equivalence can always be chosen to respect character degrees, in either of two tiers: (1) whether for every such $B$ the derived equivalence predicted by Broué's conjecture can be chosen so that the bijection induced by its perfect isometry satisfies $f(\chi)(1)\le\chi(1)$ for all $\chi\in\mathrm{Irr}(B)$; and (2) whether the equivalence can be chosen, as suggested by Theorem 4.1 of Linckelmann (2025), so that the stronger source-algebra inequality $f(\chi)(1)\le\chi(i)$ holds for all $\chi\in\mathrm{Irr}(B)$, where $i$ is a source idempotent of $B$ and $\chi(i)=\dim_K(iX)$ is the dimension of the simple $K\otimes_{\mathcal{O}}iBi$-module corresponding to $\chi$. Since $\chi(i)\le\chi(1)$, tier (2) is the stronger statement and implies tier (1). A counterexample to tier (1) is a block $B$ with abelian defect group such that no derived equivalence between $B$ and its Brauer correspondent can be chosen whose induced perfect isometry has a character bijection satisfying the tier-(1) inequality $f(\chi)(1)\le\chi(1)$ for all $\chi\in\mathrm{Irr}(B)$. A positive answer in tier (1) entails Broué's Abelian Defect Group Conjecture itself together with Giannelli's Conjecture B for all blocks with abelian defect groups.

References

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