07-1352/Class Notes for March 27

• PBW for ${\mathcal A}(\uparrow)$.
• PBW for ${\mathcal A}(\uparrow_n)$.
• The VS structure on ${\mathcal A}(\uparrow_\star)$ and on ${\mathcal B}_\star$.
• The algebra structure on ${\mathcal A}$ and ${\mathcal B}$.
• The graded representations ${\mathcal E}_k$ and their decompositions into irreducibles:
• Coloured stickers and tensor products over $S_k$.