# Many Zeros of Many Characters of GL(n,q)

@article{Gallagher2019ManyZO, title={Many Zeros of Many Characters of GL(n,q)}, author={Patrick X. Gallagher and Michael Larsen and Alexander R. Miller}, journal={International Mathematics Research Notices}, year={2019} }

For $G=\textrm{GL}(n,q)$, the proportion $P_{n,q}$ of pairs $(\chi ,g)$ in $\textrm{Irr}(G)\times G$ with $\chi (g)\neq 0$ satisfies $P_{n,q}\to 0$ as $n\to \infty $.

## 3 Citations

### Zeros and roots of unity in character tables

- Mathematics
- 2020

. For any ﬁnite group G , Thompson proved that, for each χ ∈ Irr( G ), χ ( g ) is a root of unity or zero for more than a third of the elements g ∈ G , and Gallagher proved that, for each larger than…

### Groups with a small average number of zeros in the character table

- MathematicsJournal of Algebra
- 2022

### The sparsity of character tables of high rank groups of Lie type

- MathematicsRepresentation Theory of the American Mathematical Society
- 2020

In the high rank limit, the fraction of non-zero character table entries of finite simple groups of Lie type goes to zero.

## References

SHOWING 1-8 OF 8 REFERENCES

### The probability that a character value is zero for the symmetric group

- Mathematics
- 2013

We consider random character values X(g) of the symmetric group on n symbols, where X is chosen at random from the set of irreducible characters and g is chosen at random from the group, and we show…

### The characters of the finite general linear groups

- Mathematics
- 1955

Introduction. In this paper we show how to calculate the irreducible characters of the group GL(n, q) of all nonsingular matrices of degree n with coefficients in the finite field of q elements.…

### On the Distribution of Values of Certain Word Maps

- Mathematics
- 2013

For any positive integers m and n, the word map (x,y) -> x^m y^n is almost measure preserving on large finite simple groups G.

### The distribution of the number of summands in the partitions of a positive integer

- Mathematics
- 1941

It is easily seen that the number of partitions of n having k or less summands is equal to the number of partitions of n in which no summand exceeds k . Thus the preceding results can be applied to…

### Degrees, class sizes and divisors of character values

- Mathematics
- 2012

Abstract. In the character table of a finite group there is a tendency either for the character degree to divide the conjugacy class size or the character value to vanish. There is also a partial…

### Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements

- Mathematics
- 2009

We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions…

### Graduate Texts in Mathematics

- Education
- 1982

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully…

### Sur la distribution des cycles dans les permutations

- C. R. (Doklady) Acad. Sci. URSS (N.S.)
- 1942