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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Small values and forbidden values for the Fourier anti-diagonal constant of a finite group
AU - Choi, Yemon
PY - 2025/5/19
Y1 - 2025/5/19
N2 - For a finite group G, let AD(G) denote the Fourier norm of the antidiagonal in G × G. The author showed recently that AD(G) coincides with the amenability constant of the Fourier algebra of G and is equal to the normalized sum of the cubes of the character degrees of G. Motivated by a gap result for amenability constants by B. E. Johnson, we determine exactly which numbers in the interval [1,2] arise as values of AD(G). As a by-product, we show that the set of values of AD(G) does not contain all its limit points.Some other calculations or bounds for AD(G) are given for familiar classes of finite groups. We also indicate a connection between AD(G) and the commuting probability of G, and use this to show that every finite group G satisfying AD(G) < 61/15 must be solvable; here, the value 61/15 is the best possible.
AB - For a finite group G, let AD(G) denote the Fourier norm of the antidiagonal in G × G. The author showed recently that AD(G) coincides with the amenability constant of the Fourier algebra of G and is equal to the normalized sum of the cubes of the character degrees of G. Motivated by a gap result for amenability constants by B. E. Johnson, we determine exactly which numbers in the interval [1,2] arise as values of AD(G). As a by-product, we show that the set of values of AD(G) does not contain all its limit points.Some other calculations or bounds for AD(G) are given for familiar classes of finite groups. We also indicate a connection between AD(G) and the commuting probability of G, and use this to show that every finite group G satisfying AD(G) < 61/15 must be solvable; here, the value 61/15 is the best possible.
KW - amenability constant
KW - character theory
KW - finite group
U2 - 10.1017/S1446788724000211
DO - 10.1017/S1446788724000211
M3 - Journal article
VL - 118
SP - 297
EP - 316
JO - Journal of the Australian Mathematical Society
JF - Journal of the Australian Mathematical Society
SN - 1446-7887
IS - 3
ER -