Schur’s lemma - 2026 ACF Regionals

Question

The formulator of this statement used it to prove his namesake orthogonality relations, which makes this lemma a basic tenet of representation theory. For 10 points each:

[10h] Name this statement that implies that linear maps between certain finite representations of a group that commute with the action of the group are either isomorphisms or trivial.

ANSWER: Schur’s lemma

[10m] Schur’s lemma specifically applies to representations described by this adjective. Polynomials described by this adjective can not be factored into polynomials of a lower degree.

ANSWER: irreducible [accept irreducible representations or irreps; accept irreducible polynomials; accept not reducible]

[10e] The standard proof of Schur’s lemma begins by showing that the kernel of a linear map must be zero or the entirety of this set. A function maps this set to its range.

ANSWER: domain [prompt on inputs]

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Summary


California (South)	Main Site	Y	4	20.00	100%	75%	25%
Canada	Main Site	Y	10	17.00	90%	60%	20%
Lower Mid-Atlantic	Main Site	Y	1	10.00	100%	0%	0%
Lower Mid-Atlantic	Main Site	Y	6	11.67	100%	17%	0%
Midwest	Main Site	Y	8	18.75	100%	63%	25%
Northeast	Main Site	Y	5	14.00	100%	40%	0%
Overflow	Main Site	Y	3	13.33	100%	33%	0%
UK	Main Site	Y	12	17.50	92%	67%	17%

Data


Arizona State	UC Santa Barbara	10	10	10	30
UC Irvine	UCSD C	0	0	10	10
UCLA A	UCSD B	0	10	10	20
UCLA B	UCSD A	0	10	10	20
Toronto Metropolitan	Carleton University	0	0	0	0
McMaster A	Guelph	0	0	10	10
Toronto B	Queen's	0	10	10	20
Brock	Toronto D	0	0	10	10
Simon Fraser	Toronto E	10	10	10	30
Waterloo B	Toronto F	0	10	10	20
McGill B	UBC A	0	0	10	10
NC State B	Liberty B	0	0	10	10
North Carolina A	Liberty C	0	0	10	10
North Carolina C	Liberty A	0	0	10	10
South Carolina A	VCU	0	10	10	20
South Carolina B	North Carolina B	0	0	10	10
Virginia	Virginia Tech A	0	0	10	10
William and Mary	Virginia Tech B	0	0	10	10
Ottawa	UBC B	10	10	10	30
Waterloo A	McGill A	0	10	10	20
Toronto G	Western B	0	10	10	20
Chicago D	Chicago B	10	10	10	30
Illinois A	Illinois C	0	10	10	20
Indiana A	Northwestern B	10	10	10	30
Northwestern C	Purdue B	0	0	10	10
Northwestern D	Notre Dame B	0	0	10	10
Purdue A	Illinois B	0	10	10	20
Chicago C	Chicago A	0	10	10	20
Harvard A	Bowdoin A	0	10	10	20
Southern Connecticut State	Bowdoin B	0	0	10	10
Brandeis A	Harvard B	0	0	10	10
Brandeis B	MIT	0	10	10	20
Yale	Brown	0	0	10	10
Carleton College	Iowa State	0	10	10	20
Missouri A	Georgia Tech A	0	0	10	10
Washington A	Claremont	0	0	10	10
Oxford B	Southampton B	0	10	10	20
Oxford A	Warwick A	10	10	10	30
Warwick B	City St. George's	0	10	10	20
LSE	Birmingham	0	0	10	10
Imperial A	Cambridge A	0	10	10	20
Cambridge B	Manchester	0	0	10	10
Cambridge C	Oxford C	0	10	10	20
Durham	Imperial B	0	10	10	20
Edinburgh A	Bristol	0	10	10	20
Southampton A	Edinburgh B	0	10	10	20
Cardiff	Imperial C	0	0	0	0
UCL	Montpellier	10	0	10	20
Indiana B	Chicago E	0	0	10	10