Spin characters of the symmetric group which are proportional to linear characters in characteristic 2

TL;DR

The paper completely classifies when the $2$-modular reduction of an irreducible spin representation of the double cover of the symmetric group is proportional to the Brauer character of a Specht module. Central to the result is the notion of $4$-stepped-and-semi-congruent$ tight described partitions for which a corresponding ${oldsymbol{eta}}_igcirc$ exists; the main theorem states that proportionality occurs if and only if the spin label is $4$-stepped-and-semi-congruent and the linear label is ${oldsymbol{eta}}_igcirc$ or its conjugate, with the proportionality constant depending on the number of even parts. The authors develop a suite of tools—runner-swapping and quotient-redistributing functors, abacus/quotient machinery, and detailed block analysis (RoCK blocks)—to propagate proportionality from homogeneous spin cases to general cases. This work advances modular spin representation theory by connecting abacus combinatorics, $2$-core/quotient theory, and block structure to a precise proportionality criterion, with potential implications for understanding decomposition matrices and related invariants in characteristic $2$.

Abstract

For a finite group, it is interesting to determine when two ordinary irreducible representations have the same $p$-modular reduction; that is, when two rows of the decomposition matrix in characteristic $p$ are equal, or equivalently when the corresponding $p$-modular Brauer characters are the same. We complete this task for the double covers of the symmetric group when $p=2$, by determining when the $2$-modular reduction of an irreducible spin representation coincides with a $2$-modular Specht module. In fact, we obtain a more general result: we determine when an irreducible spin representation has $2$-modular Brauer character proportional to that of a Specht module. In the course of the proof, we use induction and restriction functors to construct a function on generalised characters which has the effect of swapping runners in abacus displays for the labelling partitions.

Figures (4)

• Figure 1: An illustration of the ladder considered in \ref{['lemma:first_ladder_still_hit_going_down']} hitting the following three rows. The dashed boxes indicate nodes that may or may not be present in the Young diagram.
• Figure 2: An illustration of the largest ladder meeting row $2i-1$ or $2i$, considered in \ref{['lemma:more_ladders_hit_going_down']}, being met and possibly exceeded by row $2i+1$ or $2i+2$. The dashed boxes indicate nodes that may or may not be present in the Young diagram.
• Figure 3: The RoCK partition $(20,19,16,13,10,9,8,7,6,5,4,3,2^5,1^3)$, depicted as a Young diagram, on an abacus, and in terms of its $2$-core and $2$-quotient. Observe the correspondence between nodes in the components of the $2$-quotient and "dominoes" in the Young diagram indicated by the colouring.
• Figure 4: A comparison of the calculations of $B_{\eta,\theta}$ and $B_{\check{\eta},\check{\theta}}$ when $\eta=(7,6,2,1)$ and $\theta=(6,5,3,1)$ (and hence $\check{\eta}=(6,5,1)$ and $\check{\theta}=(5,4,2)$). This example falls into the final case ($\Delta_{1} = \Delta_{2} = 0$ and $\eta'_1 - \eta'_2 = 1$) in the proof of \ref{['intermind']}. On the left are the partitions in $\mathcal{I}_{0}(\eta,\theta)$, on the right the partitions in $\mathcal{I}_{0}(\check{\eta},\check{\theta})$, matched up according to the function $\zeta\mapsto\check{\zeta}$. The Young diagrams of the partitions are drawn, with the nodes which must be added to form $\eta$ or $\check{\eta}$ shaded blue and the nodes which must be added to form $\theta$ or $\check{\theta}$ shaded yellow (and the nodes which must be added in either case shaded green).

Theorems & Definitions (58)

• lemma 1
• lemma 2
• proof
• lemma 3
• proof
• proof
• proof
• lemma 4
• proof
• lemma 5