Exams

This page gives basic information about what is to be covered on the exams and their format. For grading details see the Grading Policies information page.

The exams are closed book exams. No calculators are permitted. You are allowed one page (8 1/2 inch by 11 inch) of notes (front and back). The exams are 1 hour and 20 minutes in length and will be during the regularly scheduled lecture time. Scratch paper will be provided. Bring (multiple) pens or pencils to write with.

Exam 1: Wednesday, March 19th 2:35 - 3:55pm, 32-141 (regular classroom and class time).

Resources:

• All the readings listed on the course calendar for Lectures 1-7.
• This website that has tabulated character tables and other useful information for finite groups

Things that you are expected to know / be able to do for Exam 1:

• Understand definitions:
  • group, subgroup, self-congjugate (normal) subgroup, order of a group, conjugacy classes, left coset, right coset, isomorphism between two groups, homomorphism between two groups, abelian vs. non-abelian groups (Lectures 2-3)
  • group representation (matrices + vectorspace), reducible representation, irreducible representation, (left and right) regular representation of a finite group, character of a representation
• Understand results of Lemmas and Theorems proved / shown in class:
  • Rearrangement Theorem
  • Lagrange's Theorem
  • Schur's Lemma
  • The Wonderful Orthogonality Theorem for Representations
  • The Wonderful Orthogonality Theorem for Character
• Know when to use and how to interpret outputs of the functions you've coded in exercsies from the symm4ml class repository
  • First exercise: groups module
  • Second and third exercises: functions from linalg and rep modules
  • Fourth exercise: vib_modes module
• Understand and be able to use the general procedure for finding vibrational modes.

Things that you may be expected to do for Exam 1:

• Generate a group from a subset of elements
• Difference between abelian vs. non-abelian groups, and classifying group based on multiplication table
• Making, completing, and identifying errors in multiplication and character tables
• Use results of Lemmas and Theorems to classify nature of representations
• Interpret the output of the functions you have code in the first 3 exercises in this course (groups, linalg, rep, vib_modes)
• Construct the Left and Right Regular Representations from the multiplication table of a group
• Decompose representation into irreps of a given group given characters of the representation.
• Use the Wonderful Orthogonality Theorem for Character and basic facts of representations to complete an incomplete character table.
• Compute the irreps of possible vibrational modes "by hand" if given a character table and a (simple) set of points that have the symmetry of that character table.
• Identify possible irreps that could lower the symmetry of a system from a group to a given subgroup.
• Given a step in a proof, identify what property of groups / representations or lemma / theorem is being used to e.g. equate the left and right hand sides of an equation.
• Use of np.einsum, broadcasting, and reshape to translate equations into code.
  • There are exercises for np.einsum and Broadcasting here under the 2. Einsum and 3. Broadcasting headings, respectively.

Exam 2: This page will be updated closer to exam dates.

Exam 3: This page will be updated closer to exam dates.