Isaacs-Navarro conjecture

The Isaacs-Navarro conjecture is a slight generalization of the McKay conjecture and is believed to hold for all finite groups. The conjecture was introduced in a 2002 paper by Isaacs and Navarro.

Statement

Suppose $G$ is a finite group and $p$ is a prime number. Denote by $f(G,p,a)$ the number of equivalence classes of irreducible representations of $G$ over the complex numbers whose degree is congruent to $a$ or $-a$ modulo $p$. Then, if $a$ is not divisible by $p$, and $P$ is a $p$-Sylow subgroup of $G$, we have:

$\!f(G,p,a)=f(N_{G}(P),p,a)$

Current status

The conjecture for all finite groups is open, but it has been resolved for some types of groups.

Group property or group family	Status of the conjecture	Explanation
finite nilpotent group	resolved	obvious: the normalizer of any Sylow subgroup is the whole group.
symmetric group on finite set	resolved	See Fong's 2003 paper on the conjecture.
alternating group on finite set	resolved	See Nath's 2009 paper on the conjecture.
double cover of symmetric group and double cover of alternating group	resolved in odd characteristic	See Gramain's 2011 paper on the conjecture.

References

• New refinements of the McKay conjecture for finite groups by I. Martin Isaacs and Gabriel Navarro, Annals of Mathematics, Volume 156, Page 333 - 344(Year 2002): ^{ArXiV copyMore info}
• The Isaacs–Navarro conjecture for symmetric groups by Paul Fong, Journal of Algebra, ISSN 00218693, Volume 260,Number 1, Page 154 - 161(February 2003): ^{Official copy (gated)More info}
• The Isaacs–Navarro conjecture for the alternating groups by Rishi Nath, Journal of Algebra, ISSN 00218693, Volume 321,Number 6, Page 1632 - 1642(March 2009): ^{Official copy (gated)More info}
• The Isaacs–Navarro conjecture for covering groups of the symmetric and alternating groups in odd characteristic by Jean-Baptiste Gramain, Journal of Algebraic Combinatorics, Volume 34,Number 3, Page 401 - 426(November 2011): ^{Official copy (gated)More info}