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Jan 12, 2024 177 likes •635 views

t trs rs rs t rt s rs rs s

SLIDE 1

✶✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s

◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

❙③❛❜♦❧❝s ▼és③ár♦s

❈❡♥tr❛❧ ❊✉r♦♣❡❛♥ ❯♥✐✈❡rs✐t②✱ ❇✉❞❛♣❡st✱ ❍✉♥❣❛r②

✷✶t❤ ❏✉♥❡✱ ✷✵✶✼

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 2

✷✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

◗✉❛♥t✉♠ ▼❛tr✐❝❡s

❉❡✜♥✐t✐♦♥ ▲❡t q ∈ k× ❛♥❞ Oq(MN) := Ctij | ✶ ≤ i,j ≤ N/(Rel) ✇❤❡r❡ Rel✿ tiktil −qtiltik tiktjk −qtjktik tjktil −tiltjk tiktjl −tjltik −(q −q−✶)tiltjk (∀i < j, k < l)         ✳ ✳ ✳ ✳ ✳ ✳ ... tik → til ... ↓

... tjk → tjl ... ✳ ✳ ✳ ✳ ✳ ✳         ∃ ˆ R ∈ CN✷×N✷ s✉❝❤ t❤❛t ˆ R s❛t✐s✜❡s t❤❡ ❨❛♥❣✲❇❛①t❡r ❊q✳ ❛♥❞ (Rel) =

k,l

ˆ Rij

kltkmtln −tiktjl ˆ

Rkl

mn | i,j,m,n ≤ N

❙③❛❜♦❧❝s ▼és③ár♦s

◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 3

✷✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

◗✉❛♥t✉♠ ▼❛tr✐❝❡s

❉❡✜♥✐t✐♦♥ ▲❡t q ∈ k× ❛♥❞ Oq(MN) := Ctij | ✶ ≤ i,j ≤ N/(Rel) ✇❤❡r❡ Rel✿ tiktil −qtiltik tiktjk −qtjktik tjktil −tiltjk tiktjl −tjltik −(q −q−✶)tiltjk (∀i < j, k < l)         ✳ ✳ ✳ ✳ ✳ ✳ ... tik → til ... ↓

... tjk → tjl ... ✳ ✳ ✳ ✳ ✳ ✳         ∃ ˆ R ∈ CN✷×N✷ s✉❝❤ t❤❛t ˆ R s❛t✐s✜❡s t❤❡ ❨❛♥❣✲❇❛①t❡r ❊q✳ ❛♥❞ (Rel) =

k,l

ˆ Rij

kltkmtln −tiktjl ˆ

Rkl

mn | i,j,m,n ≤ N

❙③❛❜♦❧❝s ▼és③ár♦s

◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 4

✷✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

◗✉❛♥t✉♠ ▼❛tr✐❝❡s

❉❡✜♥✐t✐♦♥ ▲❡t q ∈ k× ❛♥❞ Oq(MN) := Ctij | ✶ ≤ i,j ≤ N/(Rel) ✇❤❡r❡ Rel✿ tiktil −qtiltik tiktjk −qtjktik tjktil −tiltjk tiktjl −tjltik −(q −q−✶)tiltjk (∀i < j, k < l)         ✳ ✳ ✳ ✳ ✳ ✳ ... tik → til ... ↓

... tjk → tjl ... ✳ ✳ ✳ ✳ ✳ ✳         ∃ ˆ R ∈ CN✷×N✷ s✉❝❤ t❤❛t ˆ R s❛t✐s✜❡s t❤❡ ❨❛♥❣✲❇❛①t❡r ❊q✳ ❛♥❞ (Rel) =

k,l

ˆ Rij

kltkmtln −tiktjl ˆ

Rkl

mn | i,j,m,n ≤ N

❙③❛❜♦❧❝s ▼és③ár♦s

◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 5

✸✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

Pr♦♣❡rt✐❡s ♦❢ Oq(MN)

❆ss✉♠♣t✐♦♥✿ q ∈ k× ✐s ♥♦t ❛ r♦♦t ♦❢ ✉♥✐t②✳ Oq(MN) ✐s ❛ ❜✐❛❧❣❡❜r❛ ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✱ s✳t✳ ✐t ✐s ❝♦s❡♠✐s✐♠♣❧❡✱ ❞✐♠❡♥s✐♦♥s ♦❢ ✐ts s✐♠♣❧❡ ❝♦♠♦❞✉❧❡s ❛r❡ t❤❡ s❛♠❡ ❛s ♦❢ O(MN)✱ ✭❉♦♠♦❦♦s✱ ▲❡♥❛❣❛♥✱ ✷✵✵✸✮ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ r✐♥❣ ♦❢ ✐ts ❝♦♠♦❞✉❧❡s✱ t❤❛t ✐s Oq(MN)coc✱ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ♦❢ r❛♥❦ N✱ ✭❙③✳ ▼✳✱ ✷✵✶✺✮ Oq(MN)coc ✐s ❛ ♠❛①✐♠❛❧ ❝♦♠♠✉t❛t✐✈❡ s✉❜❛❧❣❡❜r❛✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 6

✸✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

Pr♦♣❡rt✐❡s ♦❢ Oq(MN)

❆ss✉♠♣t✐♦♥✿ q ∈ k× ✐s ♥♦t ❛ r♦♦t ♦❢ ✉♥✐t②✳ Oq(MN) ✐s ❛ ❜✐❛❧❣❡❜r❛ ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✱ s✳t✳ ✐t ✐s ❝♦s❡♠✐s✐♠♣❧❡✱ ❞✐♠❡♥s✐♦♥s ♦❢ ✐ts s✐♠♣❧❡ ❝♦♠♦❞✉❧❡s ❛r❡ t❤❡ s❛♠❡ ❛s ♦❢ O(MN)✱ ✭❉♦♠♦❦♦s✱ ▲❡♥❛❣❛♥✱ ✷✵✵✸✮ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ r✐♥❣ ♦❢ ✐ts ❝♦♠♦❞✉❧❡s✱ t❤❛t ✐s Oq(MN)coc✱ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ♦❢ r❛♥❦ N✱ ✭❙③✳ ▼✳✱ ✷✵✶✺✮ Oq(MN)coc ✐s ❛ ♠❛①✐♠❛❧ ❝♦♠♠✉t❛t✐✈❡ s✉❜❛❧❣❡❜r❛✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 7

✸✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

Pr♦♣❡rt✐❡s ♦❢ Oq(MN)

❆ss✉♠♣t✐♦♥✿ q ∈ k× ✐s ♥♦t ❛ r♦♦t ♦❢ ✉♥✐t②✳ Oq(MN) ✐s ❛ ❜✐❛❧❣❡❜r❛ ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✱ s✳t✳ ✐t ✐s ❝♦s❡♠✐s✐♠♣❧❡✱ ❞✐♠❡♥s✐♦♥s ♦❢ ✐ts s✐♠♣❧❡ ❝♦♠♦❞✉❧❡s ❛r❡ t❤❡ s❛♠❡ ❛s ♦❢ O(MN)✱ ✭❉♦♠♦❦♦s✱ ▲❡♥❛❣❛♥✱ ✷✵✵✸✮ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ r✐♥❣ ♦❢ ✐ts ❝♦♠♦❞✉❧❡s✱ t❤❛t ✐s Oq(MN)coc✱ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ♦❢ r❛♥❦ N✱ ✭❙③✳ ▼✳✱ ✷✵✶✺✮ Oq(MN)coc ✐s ❛ ♠❛①✐♠❛❧ ❝♦♠♠✉t❛t✐✈❡ s✉❜❛❧❣❡❜r❛✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 8

✸✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

Pr♦♣❡rt✐❡s ♦❢ Oq(MN)

❆ss✉♠♣t✐♦♥✿ q ∈ k× ✐s ♥♦t ❛ r♦♦t ♦❢ ✉♥✐t②✳ Oq(MN) ✐s ❛ ❜✐❛❧❣❡❜r❛ ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✱ s✳t✳ ✐t ✐s ❝♦s❡♠✐s✐♠♣❧❡✱ ❞✐♠❡♥s✐♦♥s ♦❢ ✐ts s✐♠♣❧❡ ❝♦♠♦❞✉❧❡s ❛r❡ t❤❡ s❛♠❡ ❛s ♦❢ O(MN)✱ ✭❉♦♠♦❦♦s✱ ▲❡♥❛❣❛♥✱ ✷✵✵✸✮ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ r✐♥❣ ♦❢ ✐ts ❝♦♠♦❞✉❧❡s✱ t❤❛t ✐s Oq(MN)coc✱ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ♦❢ r❛♥❦ N✱ ✭❙③✳ ▼✳✱ ✷✵✶✺✮ Oq(MN)coc ✐s ❛ ♠❛①✐♠❛❧ ❝♦♠♠✉t❛t✐✈❡ s✉❜❛❧❣❡❜r❛✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 9

✸✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

Pr♦♣❡rt✐❡s ♦❢ Oq(MN)

❆ss✉♠♣t✐♦♥✿ q ∈ k× ✐s ♥♦t ❛ r♦♦t ♦❢ ✉♥✐t②✳ Oq(MN) ✐s ❛ ❜✐❛❧❣❡❜r❛ ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✱ s✳t✳ ✐t ✐s ❝♦s❡♠✐s✐♠♣❧❡✱ ❞✐♠❡♥s✐♦♥s ♦❢ ✐ts s✐♠♣❧❡ ❝♦♠♦❞✉❧❡s ❛r❡ t❤❡ s❛♠❡ ❛s ♦❢ O(MN)✱ ✭❉♦♠♦❦♦s✱ ▲❡♥❛❣❛♥✱ ✷✵✵✸✮ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ r✐♥❣ ♦❢ ✐ts ❝♦♠♦❞✉❧❡s✱ t❤❛t ✐s Oq(MN)coc✱ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ♦❢ r❛♥❦ N✱ ✭❙③✳ ▼✳✱ ✷✵✶✺✮ Oq(MN)coc ✐s ❛ ♠❛①✐♠❛❧ ❝♦♠♠✉t❛t✐✈❡ s✉❜❛❧❣❡❜r❛✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 10

✸✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

Pr♦♣❡rt✐❡s ♦❢ Oq(MN)

❆ss✉♠♣t✐♦♥✿ q ∈ k× ✐s ♥♦t ❛ r♦♦t ♦❢ ✉♥✐t②✳ Oq(MN) ✐s ❛ ❜✐❛❧❣❡❜r❛ ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✱ s✳t✳ ✐t ✐s ❝♦s❡♠✐s✐♠♣❧❡✱ ❞✐♠❡♥s✐♦♥s ♦❢ ✐ts s✐♠♣❧❡ ❝♦♠♦❞✉❧❡s ❛r❡ t❤❡ s❛♠❡ ❛s ♦❢ O(MN)✱ ✭❉♦♠♦❦♦s✱ ▲❡♥❛❣❛♥✱ ✷✵✵✸✮ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ r✐♥❣ ♦❢ ✐ts ❝♦♠♦❞✉❧❡s✱ t❤❛t ✐s Oq(MN)coc✱ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ♦❢ r❛♥❦ N✱ ✭❙③✳ ▼✳✱ ✷✵✶✺✮ Oq(MN)coc ✐s ❛ ♠❛①✐♠❛❧ ❝♦♠♠✉t❛t✐✈❡ s✉❜❛❧❣❡❜r❛✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 11

✹✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

❘✐♥❣ t❤❡♦r② ♦❢ Oq(MN)

Oq(MN) ✐s ❛ ✜♥✳ ♣r❡s✳ ❣r❛❞❡❞ ❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ✐♥ ❞❡❣r❡❡ ✶✱ s✳t✳ ✐t ✐s ❛ ◆♦❡t❤❡r✐❛♥ ❞♦♠❛✐♥✱ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❯❋❉✱ ❝❛t❡♥❛r②✱ ❡t❝✳✱

✶ ✐t ✐s ❛ P❇❲✲❛❧❣❡❜r❛✱ ✐✳❡✳ ♦r❞❡r❡❞ ♠♦♥♦♠✐❛❧s ✐♥ t✶✶,t✶✷,...,tnn

❢♦r♠ ❛ ❜❛s✐s ✐♥ Oq(MN)✱

✷ ✐ts r❡❧❛t✐♦♥s ❛r❡ ♦❢ t❤❡ ❢♦r♠ ( ˆ

RTT −TT ˆ R) ❢♦r s♦♠❡ ˆ R s✳t✳ Yang-Baxter Eq.: ˆ R✶✷ ˆ R✷✸ ˆ R✶✷ = ˆ R✷✸ ˆ R✶✷ ˆ R✷✸ Hecke equation: ( ˆ R +✶)( ˆ R −q) = ✵ T (V )/Im( ˆ R +✶) ❛♥❞ T (V )/Im( ˆ R −q) ❛r❡ P❇❲✲❛❧❣❡❜r❛s✳ ❚❤❡♦r❡♠ ✭❙✉❞❜❡r②✮ ✷✮ ⇒ ✶✮ ✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 12

✹✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

❘✐♥❣ t❤❡♦r② ♦❢ Oq(MN)

Oq(MN) ✐s ❛ ✜♥✳ ♣r❡s✳ ❣r❛❞❡❞ ❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ✐♥ ❞❡❣r❡❡ ✶✱ s✳t✳ ✐t ✐s ❛ ◆♦❡t❤❡r✐❛♥ ❞♦♠❛✐♥✱ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❯❋❉✱ ❝❛t❡♥❛r②✱ ❡t❝✳✱

✶ ✐t ✐s ❛ P❇❲✲❛❧❣❡❜r❛✱ ✐✳❡✳ ♦r❞❡r❡❞ ♠♦♥♦♠✐❛❧s ✐♥ t✶✶,t✶✷,...,tnn

❢♦r♠ ❛ ❜❛s✐s ✐♥ Oq(MN)✱

✷ ✐ts r❡❧❛t✐♦♥s ❛r❡ ♦❢ t❤❡ ❢♦r♠ ( ˆ

RTT −TT ˆ R) ❢♦r s♦♠❡ ˆ R s✳t✳ Yang-Baxter Eq.: ˆ R✶✷ ˆ R✷✸ ˆ R✶✷ = ˆ R✷✸ ˆ R✶✷ ˆ R✷✸ Hecke equation: ( ˆ R +✶)( ˆ R −q) = ✵ T (V )/Im( ˆ R +✶) ❛♥❞ T (V )/Im( ˆ R −q) ❛r❡ P❇❲✲❛❧❣❡❜r❛s✳ ❚❤❡♦r❡♠ ✭❙✉❞❜❡r②✮ ✷✮ ⇒ ✶✮ ✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 13

✹✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

❘✐♥❣ t❤❡♦r② ♦❢ Oq(MN)

Oq(MN) ✐s ❛ ✜♥✳ ♣r❡s✳ ❣r❛❞❡❞ ❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ✐♥ ❞❡❣r❡❡ ✶✱ s✳t✳ ✐t ✐s ❛ ◆♦❡t❤❡r✐❛♥ ❞♦♠❛✐♥✱ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❯❋❉✱ ❝❛t❡♥❛r②✱ ❡t❝✳✱

✶ ✐t ✐s ❛ P❇❲✲❛❧❣❡❜r❛✱ ✐✳❡✳ ♦r❞❡r❡❞ ♠♦♥♦♠✐❛❧s ✐♥ t✶✶,t✶✷,...,tnn

❢♦r♠ ❛ ❜❛s✐s ✐♥ Oq(MN)✱

✷ ✐ts r❡❧❛t✐♦♥s ❛r❡ ♦❢ t❤❡ ❢♦r♠ ( ˆ

RTT −TT ˆ R) ❢♦r s♦♠❡ ˆ R s✳t✳ Yang-Baxter Eq.: ˆ R✶✷ ˆ R✷✸ ˆ R✶✷ = ˆ R✷✸ ˆ R✶✷ ˆ R✷✸ Hecke equation: ( ˆ R +✶)( ˆ R −q) = ✵ T (V )/Im( ˆ R +✶) ❛♥❞ T (V )/Im( ˆ R −q) ❛r❡ P❇❲✲❛❧❣❡❜r❛s✳ ❚❤❡♦r❡♠ ✭❙✉❞❜❡r②✮ ✷✮ ⇒ ✶✮ ✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 14

✹✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

❘✐♥❣ t❤❡♦r② ♦❢ Oq(MN)

Oq(MN) ✐s ❛ ✜♥✳ ♣r❡s✳ ❣r❛❞❡❞ ❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ✐♥ ❞❡❣r❡❡ ✶✱ s✳t✳ ✐t ✐s ❛ ◆♦❡t❤❡r✐❛♥ ❞♦♠❛✐♥✱ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❯❋❉✱ ❝❛t❡♥❛r②✱ ❡t❝✳✱

✶ ✐t ✐s ❛ P❇❲✲❛❧❣❡❜r❛✱ ✐✳❡✳ ♦r❞❡r❡❞ ♠♦♥♦♠✐❛❧s ✐♥ t✶✶,t✶✷,...,tnn

❢♦r♠ ❛ ❜❛s✐s ✐♥ Oq(MN)✱

✷ ✐ts r❡❧❛t✐♦♥s ❛r❡ ♦❢ t❤❡ ❢♦r♠ ( ˆ

RTT −TT ˆ R) ❢♦r s♦♠❡ ˆ R s✳t✳ Yang-Baxter Eq.: ˆ R✶✷ ˆ R✷✸ ˆ R✶✷ = ˆ R✷✸ ˆ R✶✷ ˆ R✷✸ Hecke equation: ( ˆ R +✶)( ˆ R −q) = ✵ T (V )/Im( ˆ R +✶) ❛♥❞ T (V )/Im( ˆ R −q) ❛r❡ P❇❲✲❛❧❣❡❜r❛s✳ ❚❤❡♦r❡♠ ✭❙✉❞❜❡r②✮ ✷✮ ⇒ ✶✮ ✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 15

✹✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

❘✐♥❣ t❤❡♦r② ♦❢ Oq(MN)

Oq(MN) ✐s ❛ ✜♥✳ ♣r❡s✳ ❣r❛❞❡❞ ❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ✐♥ ❞❡❣r❡❡ ✶✱ s✳t✳ ✐t ✐s ❛ ◆♦❡t❤❡r✐❛♥ ❞♦♠❛✐♥✱ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❯❋❉✱ ❝❛t❡♥❛r②✱ ❡t❝✳✱

✶ ✐t ✐s ❛ P❇❲✲❛❧❣❡❜r❛✱ ✐✳❡✳ ♦r❞❡r❡❞ ♠♦♥♦♠✐❛❧s ✐♥ t✶✶,t✶✷,...,tnn

❢♦r♠ ❛ ❜❛s✐s ✐♥ Oq(MN)✱

✷ ✐ts r❡❧❛t✐♦♥s ❛r❡ ♦❢ t❤❡ ❢♦r♠ ( ˆ

RTT −TT ˆ R) ❢♦r s♦♠❡ ˆ R s✳t✳ Yang-Baxter Eq.: ˆ R✶✷ ˆ R✷✸ ˆ R✶✷ = ˆ R✷✸ ˆ R✶✷ ˆ R✷✸ Hecke equation: ( ˆ R +✶)( ˆ R −q) = ✵ T (V )/Im( ˆ R +✶) ❛♥❞ T (V )/Im( ˆ R −q) ❛r❡ P❇❲✲❛❧❣❡❜r❛s✳ ❚❤❡♦r❡♠ ✭❙✉❞❜❡r②✮ ✷✮ ⇒ ✶✮ ✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 16

✹✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s

❘✐♥❣ t❤❡♦r② ♦❢ Oq(MN)

Oq(MN) ✐s ❛ ✜♥✳ ♣r❡s✳ ❣r❛❞❡❞ ❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ✐♥ ❞❡❣r❡❡ ✶✱ s✳t✳ ✐t ✐s ❛ ◆♦❡t❤❡r✐❛♥ ❞♦♠❛✐♥✱ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ❯❋❉✱ ❝❛t❡♥❛r②✱ ❡t❝✳✱

✶ ✐t ✐s ❛ P❇❲✲❛❧❣❡❜r❛✱ ✐✳❡✳ ♦r❞❡r❡❞ ♠♦♥♦♠✐❛❧s ✐♥ t✶✶,t✶✷,...,tnn

❢♦r♠ ❛ ❜❛s✐s ✐♥ Oq(MN)✱

✷ ✐ts r❡❧❛t✐♦♥s ❛r❡ ♦❢ t❤❡ ❢♦r♠ ( ˆ

RTT −TT ˆ R) ❢♦r s♦♠❡ ˆ R s✳t✳ Yang-Baxter Eq.: ˆ R✶✷ ˆ R✷✸ ˆ R✶✷ = ˆ R✷✸ ˆ R✶✷ ˆ R✷✸ Hecke equation: ( ˆ R +✶)( ˆ R −q) = ✵ T (V )/Im( ˆ R +✶) ❛♥❞ T (V )/Im( ˆ R −q) ❛r❡ P❇❲✲❛❧❣❡❜r❛s✳ ❚❤❡♦r❡♠ ✭❙✉❞❜❡r②✮ ✷✮ ⇒ ✶✮ ✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 17

✺✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s

❉❡✜♥✐t✐♦♥✴Pr♦♣♦s✐t✐♦♥ ✭▼❛♥✐♥✱ ❚❛❦❡✉❝❤✐✱ ❙✉❞❜❡r②✱ ✳✳✳✮ ❋♦r ❛♥② ❞❡❝♦♠♣✳ V ⊗✷ = S ⊕T✱ M (S,T) := T

End(V )
/τ✷✸(S ⊗S⊥ +T ⊗T ⊥)

✐s ❛ ❜✐❛❧❣❡❜r❛ ❝♦❛❝t✐♥❣ ♦♥ T (V )/(S) ❛♥❞ T (V )/(T)✳ ❋♦r (S,T)✿ Sym✷(V ), Λ✷(V ) − → O(MN) Sym✷

q(V ), Λ✷ q(V ) −

→ Oq(MN) ✇❤❡r❡ Λ✷

q(V ) = xixj −qxjxi | ✶ ≤ i < j ≤ N

Sym✷

q(V ) = qxixj +xjxi | ✶ ≤ i < j ≤ N

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 18

✺✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s

❉❡✜♥✐t✐♦♥✴Pr♦♣♦s✐t✐♦♥ ✭▼❛♥✐♥✱ ❚❛❦❡✉❝❤✐✱ ❙✉❞❜❡r②✱ ✳✳✳✮ ❋♦r ❛♥② ❞❡❝♦♠♣✳ V ⊗✷ = S ⊕T✱ M (S,T) := T

End(V )
/τ✷✸(S ⊗S⊥ +T ⊗T ⊥)

✐s ❛ ❜✐❛❧❣❡❜r❛ ❝♦❛❝t✐♥❣ ♦♥ T (V )/(S) ❛♥❞ T (V )/(T)✳ ❋♦r (S,T)✿ Sym✷(V ), Λ✷(V ) − → O(MN) Sym✷

q(V ), Λ✷ q(V ) −

→ Oq(MN) ✇❤❡r❡ Λ✷

q(V ) = xixj −qxjxi | ✶ ≤ i < j ≤ N

Sym✷

q(V ) = qxixj +xjxi | ✶ ≤ i < j ≤ N

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 19

✺✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s

❉❡✜♥✐t✐♦♥✴Pr♦♣♦s✐t✐♦♥ ✭▼❛♥✐♥✱ ❚❛❦❡✉❝❤✐✱ ❙✉❞❜❡r②✱ ✳✳✳✮ ❋♦r ❛♥② ❞❡❝♦♠♣✳ V ⊗✷ = S ⊕T✱ M (S,T) := T

End(V )
/τ✷✸(S ⊗S⊥ +T ⊗T ⊥)

✐s ❛ ❜✐❛❧❣❡❜r❛ ❝♦❛❝t✐♥❣ ♦♥ T (V )/(S) ❛♥❞ T (V )/(T)✳ ❋♦r (S,T)✿ Sym✷(V ), Λ✷(V ) − → O(MN) Sym✷

q(V ), Λ✷ q(V ) −

→ Oq(MN) ✇❤❡r❡ Λ✷

q(V ) = xixj −qxjxi | ✶ ≤ i < j ≤ N

Sym✷

q(V ) = qxixj +xjxi | ✶ ≤ i < j ≤ N

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 20

✺✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s

❉❡✜♥✐t✐♦♥✴Pr♦♣♦s✐t✐♦♥ ✭▼❛♥✐♥✱ ❚❛❦❡✉❝❤✐✱ ❙✉❞❜❡r②✱ ✳✳✳✮ ❋♦r ❛♥② ❞❡❝♦♠♣✳ V ⊗✷ = S ⊕T✱ M (S,T) := T

End(V )
/τ✷✸(S ⊗S⊥ +T ⊗T ⊥)

✐s ❛ ❜✐❛❧❣❡❜r❛ ❝♦❛❝t✐♥❣ ♦♥ T (V )/(S) ❛♥❞ T (V )/(T)✳ ❋♦r (S,T)✿ Sym✷(V ), Λ✷(V ) − → O(MN) Sym✷

q(V ), Λ✷ q(V ) −

→ Oq(MN) ✇❤❡r❡ Λ✷

q(V ) = xixj −qxjxi | ✶ ≤ i < j ≤ N

Sym✷

q(V ) = qxixj +xjxi | ✶ ≤ i < j ≤ N

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 21

✺✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s

❉❡✜♥✐t✐♦♥✴Pr♦♣♦s✐t✐♦♥ ✭▼❛♥✐♥✱ ❚❛❦❡✉❝❤✐✱ ❙✉❞❜❡r②✱ ✳✳✳✮ ❋♦r ❛♥② ❞❡❝♦♠♣✳ V ⊗✷ = S ⊕T✱ M (S,T) := T

End(V )
/τ✷✸(S ⊗S⊥ +T ⊗T ⊥)

✐s ❛ ❜✐❛❧❣❡❜r❛ ❝♦❛❝t✐♥❣ ♦♥ T (V )/(S) ❛♥❞ T (V )/(T)✳ ❋♦r (S,T)✿ Sym✷(V ), Λ✷(V ) − → O(MN) Sym✷

q(V ), Λ✷ q(V ) −

→ Oq(MN) ✇❤❡r❡ Λ✷

q(V ) = xixj −qxjxi | ✶ ≤ i < j ≤ N

Sym✷

q(V ) = qxixj +xjxi | ✶ ≤ i < j ≤ N

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 22

✻✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ M (S,T)

M := M (S,T) ✐s ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ q✉❛❞r❛t✐❝ ❛❧❣❡❜r❛✱ ❞✐♠M✶ = N✷ ❛♥❞ ❞✐♠M✷ = (❞✐♠S)✷ +(❞✐♠T)✷✱ ❆ss✉♠❡ t❤❛t ❞✐♠S = ❞✐♠Sym✷(V ) = N +✶ ✷

❤❡♥❝❡ ❞✐♠T = ❞✐♠Λ✷(V ) =

N

✷

❛♥❞

❞✐♠M✷ = ❞✐♠O(MN)✷ = N✷+✶

✷

◗✉❡st✐♦♥ ❲❤❡♥ ❞♦❡s M ❤❛✈❡ ❛ P❇❲✲❜❛s✐s✱ ✐✳❡✳ ❛♥ ♦r❞❡r❡❞ ❜❛s✐s ✐♥ M✶ s✳t✳ ♦r❞❡r❡❞ ♠♦♥♦♠✐❛❧s ❢♦r♠ ❛ ❜❛s✐s ✐♥ M ❄ ■♥ ♣❛rt✐❝✉❧❛r✱ ❞✐♠M✸ =?

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 23

✻✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ M (S,T)

M := M (S,T) ✐s ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ q✉❛❞r❛t✐❝ ❛❧❣❡❜r❛✱ ❞✐♠M✶ = N✷ ❛♥❞ ❞✐♠M✷ = (❞✐♠S)✷ +(❞✐♠T)✷✱ ❆ss✉♠❡ t❤❛t ❞✐♠S = ❞✐♠Sym✷(V ) = N +✶ ✷

❤❡♥❝❡ ❞✐♠T = ❞✐♠Λ✷(V ) =

N

✷

❛♥❞

❞✐♠M✷ = ❞✐♠O(MN)✷ = N✷+✶

✷

◗✉❡st✐♦♥ ❲❤❡♥ ❞♦❡s M ❤❛✈❡ ❛ P❇❲✲❜❛s✐s✱ ✐✳❡✳ ❛♥ ♦r❞❡r❡❞ ❜❛s✐s ✐♥ M✶ s✳t✳ ♦r❞❡r❡❞ ♠♦♥♦♠✐❛❧s ❢♦r♠ ❛ ❜❛s✐s ✐♥ M ❄ ■♥ ♣❛rt✐❝✉❧❛r✱ ❞✐♠M✸ =?

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 24

✻✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ M (S,T)

M := M (S,T) ✐s ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ q✉❛❞r❛t✐❝ ❛❧❣❡❜r❛✱ ❞✐♠M✶ = N✷ ❛♥❞ ❞✐♠M✷ = (❞✐♠S)✷ +(❞✐♠T)✷✱ ❆ss✉♠❡ t❤❛t ❞✐♠S = ❞✐♠Sym✷(V ) = N +✶ ✷

❤❡♥❝❡ ❞✐♠T = ❞✐♠Λ✷(V ) =

N

✷

❛♥❞

❞✐♠M✷ = ❞✐♠O(MN)✷ = N✷+✶

✷

◗✉❡st✐♦♥ ❲❤❡♥ ❞♦❡s M ❤❛✈❡ ❛ P❇❲✲❜❛s✐s✱ ✐✳❡✳ ❛♥ ♦r❞❡r❡❞ ❜❛s✐s ✐♥ M✶ s✳t✳ ♦r❞❡r❡❞ ♠♦♥♦♠✐❛❧s ❢♦r♠ ❛ ❜❛s✐s ✐♥ M ❄ ■♥ ♣❛rt✐❝✉❧❛r✱ ❞✐♠M✸ =?

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 25

✻✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ M (S,T)

M := M (S,T) ✐s ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ q✉❛❞r❛t✐❝ ❛❧❣❡❜r❛✱ ❞✐♠M✶ = N✷ ❛♥❞ ❞✐♠M✷ = (❞✐♠S)✷ +(❞✐♠T)✷✱ ❆ss✉♠❡ t❤❛t ❞✐♠S = ❞✐♠Sym✷(V ) = N +✶ ✷

❤❡♥❝❡ ❞✐♠T = ❞✐♠Λ✷(V ) =

N

✷

❛♥❞

❞✐♠M✷ = ❞✐♠O(MN)✷ = N✷+✶

✷

◗✉❡st✐♦♥ ❲❤❡♥ ❞♦❡s M ❤❛✈❡ ❛ P❇❲✲❜❛s✐s✱ ✐✳❡✳ ❛♥ ♦r❞❡r❡❞ ❜❛s✐s ✐♥ M✶ s✳t✳ ♦r❞❡r❡❞ ♠♦♥♦♠✐❛❧s ❢♦r♠ ❛ ❜❛s✐s ✐♥ M ❄ ■♥ ♣❛rt✐❝✉❧❛r✱ ❞✐♠M✸ =?

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 26

✻✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ M (S,T)

M := M (S,T) ✐s ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ q✉❛❞r❛t✐❝ ❛❧❣❡❜r❛✱ ❞✐♠M✶ = N✷ ❛♥❞ ❞✐♠M✷ = (❞✐♠S)✷ +(❞✐♠T)✷✱ ❆ss✉♠❡ t❤❛t ❞✐♠S = ❞✐♠Sym✷(V ) = N +✶ ✷

❤❡♥❝❡ ❞✐♠T = ❞✐♠Λ✷(V ) =

N

✷

❛♥❞

❞✐♠M✷ = ❞✐♠O(MN)✷ = N✷+✶

✷

◗✉❡st✐♦♥ ❲❤❡♥ ❞♦❡s M ❤❛✈❡ ❛ P❇❲✲❜❛s✐s✱ ✐✳❡✳ ❛♥ ♦r❞❡r❡❞ ❜❛s✐s ✐♥ M✶ s✳t✳ ♦r❞❡r❡❞ ♠♦♥♦♠✐❛❧s ❢♦r♠ ❛ ❜❛s✐s ✐♥ M ❄ ■♥ ♣❛rt✐❝✉❧❛r✱ ❞✐♠M✸ =?

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 27

✻✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ M (S,T)

M := M (S,T) ✐s ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ q✉❛❞r❛t✐❝ ❛❧❣❡❜r❛✱ ❞✐♠M✶ = N✷ ❛♥❞ ❞✐♠M✷ = (❞✐♠S)✷ +(❞✐♠T)✷✱ ❆ss✉♠❡ t❤❛t ❞✐♠S = ❞✐♠Sym✷(V ) = N +✶ ✷

❤❡♥❝❡ ❞✐♠T = ❞✐♠Λ✷(V ) =

N

✷

❛♥❞

❞✐♠M✷ = ❞✐♠O(MN)✷ = N✷+✶

✷

◗✉❡st✐♦♥ ❲❤❡♥ ❞♦❡s M ❤❛✈❡ ❛ P❇❲✲❜❛s✐s✱ ✐✳❡✳ ❛♥ ♦r❞❡r❡❞ ❜❛s✐s ✐♥ M✶ s✳t✳ ♦r❞❡r❡❞ ♠♦♥♦♠✐❛❧s ❢♦r♠ ❛ ❜❛s✐s ✐♥ M ❄ ■♥ ♣❛rt✐❝✉❧❛r✱ ❞✐♠M✸ =?

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 28

✼✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❊①✐st❡♥❝❡ ♦❢ ❛♥ R✲♠❛tr✐①

❚❤❡♦r❡♠ ✭❙③✳ ▼✳✮

▲❡t (V ,.,.) ❜❡ ❛ ✜♥✳ ❞✐♠✳ ❍✐❧❜❡rt s♣❛❝❡✱ S ⊆ V ⊗V ✱ T = S⊥✳ ■❢ ❞✐♠(S ⊗V ∩V ⊗S) = N +✷ ✸

❞✐♠(S ⊗V ∩V ⊗T) = ✵

t❤❡♥ ❞✐♠M✸ ≤ ❞✐♠O(MN)✸ = N✷ +✷ ✸

✇✐t❤ ❡q✉❛❧✐t② ✐❢ ❛♥❞ ♦♥❧② ✐❢ Rel(M ) = ( ˆ

RTT −TT ˆ R) ❢♦r s♦♠❡ ˆ R ∈ End(V ⊗V ) s✳t✳ ˆ R✶✷ ˆ R✷✸ ˆ R✶✷ = ˆ R✷✸ ˆ R✶✷ ˆ R✷✸ ▼♦r❡♦✈❡r✱ ✐♥ t❤✐s ❝❛s❡✱ ˆ R s❛t✐s✜❡s t❤❡ ❍❡❝❦❡ ❡q✉❛t✐♦♥✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 29

✼✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❊①✐st❡♥❝❡ ♦❢ ❛♥ R✲♠❛tr✐①

❚❤❡♦r❡♠ ✭❙③✳ ▼✳✮

▲❡t (V ,.,.) ❜❡ ❛ ✜♥✳ ❞✐♠✳ ❍✐❧❜❡rt s♣❛❝❡✱ S ⊆ V ⊗V ✱ T = S⊥✳ ■❢ ❞✐♠(S ⊗V ∩V ⊗S) = N +✷ ✸

❞✐♠(S ⊗V ∩V ⊗T) = ✵

t❤❡♥ ❞✐♠M✸ ≤ ❞✐♠O(MN)✸ = N✷ +✷ ✸

✇✐t❤ ❡q✉❛❧✐t② ✐❢ ❛♥❞ ♦♥❧② ✐❢ Rel(M ) = ( ˆ

RTT −TT ˆ R) ❢♦r s♦♠❡ ˆ R ∈ End(V ⊗V ) s✳t✳ ˆ R✶✷ ˆ R✷✸ ˆ R✶✷ = ˆ R✷✸ ˆ R✶✷ ˆ R✷✸ ▼♦r❡♦✈❡r✱ ✐♥ t❤✐s ❝❛s❡✱ ˆ R s❛t✐s✜❡s t❤❡ ❍❡❝❦❡ ❡q✉❛t✐♦♥✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 30

✽✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❋♦✉r s✉❜s♣❛❝❡ q✉✐✈❡r

❖❜s❡r✈❛t✐♦♥ ✭❣❡♥❡r❛❧✐③✐♥❣ ❘❛❡❞s❝❤❡❧❞❡rs✱ ❱❛♥ ❞❡♥ ❇❡r❣❤✮ ❆s ❛♥ ❛❧❣❡❜r❛ M ∨

n ∼

=

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 31

✾✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❋♦✉r s✉❜s♣❛❝❡ q✉✐✈❡r

❖❜s❡r✈❛t✐♦♥ ✭❜❛s✐❝❛❧❧② ❜② ▼✳ ❱❛♥ ❞❡♥ ❇❡r❣❤✮ ❆s ❛♥ ❛❧❣❡❜r❛ M ∨

n ∼

=

e ∈ End(V ⊗n) | e(U) ⊆ U,

U ∈ {Si,i+✶,Ti,i+✶ | i = ✶,...,k −✶}

✇❤❡r❡ Si,i+✶ = V ⊗(i−✶) ⊗S ⊗V ⊗(n−i−✶) ❛♥❞ s✐♠✐❧❛r❧② ❢♦r T✳

❍❡♥❝❡✱ M ∨

✸ ∼

= End˜

D✹(ρ)

❢♦r t❤❡ r❡♣✳ ρ = (V ⊗✸,S✶✷,S✷✸,T✶✷,T✷✸) ♦❢ t❤❡ ✭t❛♠❡ ❊✉❝❧✐❞❡❛♥✮ q✉✐✈❡r ˜ D✹✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 32

✾✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❋♦✉r s✉❜s♣❛❝❡ q✉✐✈❡r

❖❜s❡r✈❛t✐♦♥ ✭❜❛s✐❝❛❧❧② ❜② ▼✳ ❱❛♥ ❞❡♥ ❇❡r❣❤✮ ❆s ❛♥ ❛❧❣❡❜r❛ M ∨

n ∼

=

e ∈ End(V ⊗n) | e(U) ⊆ U,

U ∈ {Si,i+✶,Ti,i+✶ | i = ✶,...,k −✶}

✇❤❡r❡ Si,i+✶ = V ⊗(i−✶) ⊗S ⊗V ⊗(n−i−✶) ❛♥❞ s✐♠✐❧❛r❧② ❢♦r T✳

❍❡♥❝❡✱ M ∨

✸ ∼

= End˜

D✹(ρ)

❢♦r t❤❡ r❡♣✳ ρ = (V ⊗✸,S✶✷,S✷✸,T✶✷,T✷✸) ♦❢ t❤❡ ✭t❛♠❡ ❊✉❝❧✐❞❡❛♥✮ q✉✐✈❡r ˜ D✹✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 33

✶✵✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❖rt❤♦❣♦♥❛❧ ❝❛s❡

❙♣❡❝✐❛❧ ❝❛s❡✿ V ✐s ❛ ✜♥✳ ❞✐♠✳ ❍✐❧❜❡rt s♣❛❝❡✱ T = S⊥✳ ❚❤❡ ❝♦♥✜❣✳ (S✶✷,S✷✸,S⊥

✶✷,S⊥ ✷✸) ✐s ❞❡t❡r♠✐♥❡❞ ❜②

Spec(ProjS✶✷ ◦ProjS✷✸ ◦ProjS✶✷) ❛s ❛ ♠✉❧t✐s❡t ✭◆❛♠❡ ✐♥ r❡❛❧ ❝❛s❡✿ ♣r✐♥❝✐♣❛❧ ❛♥❣❧❡s✳✮✳ ❋❡✇❡r ❞✐st✐♥❝t ❛♥❣❧❡s ⇒ ♠♦r❡ ❡♥❞♦♠♦r♣❤✐s♠s ♦❢ ρ✳ S✶✷ ∩S✷✸ ❣✐✈❡s ❡✐❣❡♥✈❛❧✉❡ ✶✱ S⊥

✶✷ ❣✐✈❡s ✵✱

■❢ λ,µ ∈ Spec(PS✶✷PS✷✸PS✶✷)\{✵,✶}✱ λ = µ t❤❡♥ ∄ ˆ R ❢♦r M ✳ Greatest ❞✐♠M✸ ⇐ ⇒ S✶✷ and S✷✸ are isoclinic modulo S✶✷ ∩S✷✸ ⇐ ⇒ ∃ ˆ R

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 34

✶✵✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❖rt❤♦❣♦♥❛❧ ❝❛s❡

❙♣❡❝✐❛❧ ❝❛s❡✿ V ✐s ❛ ✜♥✳ ❞✐♠✳ ❍✐❧❜❡rt s♣❛❝❡✱ T = S⊥✳ ❚❤❡ ❝♦♥✜❣✳ (S✶✷,S✷✸,S⊥

✶✷,S⊥ ✷✸) ✐s ❞❡t❡r♠✐♥❡❞ ❜②

Spec(ProjS✶✷ ◦ProjS✷✸ ◦ProjS✶✷) ❛s ❛ ♠✉❧t✐s❡t ✭◆❛♠❡ ✐♥ r❡❛❧ ❝❛s❡✿ ♣r✐♥❝✐♣❛❧ ❛♥❣❧❡s✳✮✳ ❋❡✇❡r ❞✐st✐♥❝t ❛♥❣❧❡s ⇒ ♠♦r❡ ❡♥❞♦♠♦r♣❤✐s♠s ♦❢ ρ✳ S✶✷ ∩S✷✸ ❣✐✈❡s ❡✐❣❡♥✈❛❧✉❡ ✶✱ S⊥

✶✷ ❣✐✈❡s ✵✱

■❢ λ,µ ∈ Spec(PS✶✷PS✷✸PS✶✷)\{✵,✶}✱ λ = µ t❤❡♥ ∄ ˆ R ❢♦r M ✳ Greatest ❞✐♠M✸ ⇐ ⇒ S✶✷ and S✷✸ are isoclinic modulo S✶✷ ∩S✷✸ ⇐ ⇒ ∃ ˆ R

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 35

✶✵✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❖rt❤♦❣♦♥❛❧ ❝❛s❡

❙♣❡❝✐❛❧ ❝❛s❡✿ V ✐s ❛ ✜♥✳ ❞✐♠✳ ❍✐❧❜❡rt s♣❛❝❡✱ T = S⊥✳ ❚❤❡ ❝♦♥✜❣✳ (S✶✷,S✷✸,S⊥

✶✷,S⊥ ✷✸) ✐s ❞❡t❡r♠✐♥❡❞ ❜②

Spec(ProjS✶✷ ◦ProjS✷✸ ◦ProjS✶✷) ❛s ❛ ♠✉❧t✐s❡t ✭◆❛♠❡ ✐♥ r❡❛❧ ❝❛s❡✿ ♣r✐♥❝✐♣❛❧ ❛♥❣❧❡s✳✮✳ ❋❡✇❡r ❞✐st✐♥❝t ❛♥❣❧❡s ⇒ ♠♦r❡ ❡♥❞♦♠♦r♣❤✐s♠s ♦❢ ρ✳ S✶✷ ∩S✷✸ ❣✐✈❡s ❡✐❣❡♥✈❛❧✉❡ ✶✱ S⊥

✶✷ ❣✐✈❡s ✵✱

■❢ λ,µ ∈ Spec(PS✶✷PS✷✸PS✶✷)\{✵,✶}✱ λ = µ t❤❡♥ ∄ ˆ R ❢♦r M ✳ Greatest ❞✐♠M✸ ⇐ ⇒ S✶✷ and S✷✸ are isoclinic modulo S✶✷ ∩S✷✸ ⇐ ⇒ ∃ ˆ R

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 36

✶✵✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❖rt❤♦❣♦♥❛❧ ❝❛s❡

❙♣❡❝✐❛❧ ❝❛s❡✿ V ✐s ❛ ✜♥✳ ❞✐♠✳ ❍✐❧❜❡rt s♣❛❝❡✱ T = S⊥✳ ❚❤❡ ❝♦♥✜❣✳ (S✶✷,S✷✸,S⊥

✶✷,S⊥ ✷✸) ✐s ❞❡t❡r♠✐♥❡❞ ❜②

Spec(ProjS✶✷ ◦ProjS✷✸ ◦ProjS✶✷) ❛s ❛ ♠✉❧t✐s❡t ✭◆❛♠❡ ✐♥ r❡❛❧ ❝❛s❡✿ ♣r✐♥❝✐♣❛❧ ❛♥❣❧❡s✳✮✳ ❋❡✇❡r ❞✐st✐♥❝t ❛♥❣❧❡s ⇒ ♠♦r❡ ❡♥❞♦♠♦r♣❤✐s♠s ♦❢ ρ✳ S✶✷ ∩S✷✸ ❣✐✈❡s ❡✐❣❡♥✈❛❧✉❡ ✶✱ S⊥

✶✷ ❣✐✈❡s ✵✱

■❢ λ,µ ∈ Spec(PS✶✷PS✷✸PS✶✷)\{✵,✶}✱ λ = µ t❤❡♥ ∄ ˆ R ❢♦r M ✳ Greatest ❞✐♠M✸ ⇐ ⇒ S✶✷ and S✷✸ are isoclinic modulo S✶✷ ∩S✷✸ ⇐ ⇒ ∃ ˆ R

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 37

✶✵✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❖rt❤♦❣♦♥❛❧ ❝❛s❡

❙♣❡❝✐❛❧ ❝❛s❡✿ V ✐s ❛ ✜♥✳ ❞✐♠✳ ❍✐❧❜❡rt s♣❛❝❡✱ T = S⊥✳ ❚❤❡ ❝♦♥✜❣✳ (S✶✷,S✷✸,S⊥

✶✷,S⊥ ✷✸) ✐s ❞❡t❡r♠✐♥❡❞ ❜②

Spec(ProjS✶✷ ◦ProjS✷✸ ◦ProjS✶✷) ❛s ❛ ♠✉❧t✐s❡t ✭◆❛♠❡ ✐♥ r❡❛❧ ❝❛s❡✿ ♣r✐♥❝✐♣❛❧ ❛♥❣❧❡s✳✮✳ ❋❡✇❡r ❞✐st✐♥❝t ❛♥❣❧❡s ⇒ ♠♦r❡ ❡♥❞♦♠♦r♣❤✐s♠s ♦❢ ρ✳ S✶✷ ∩S✷✸ ❣✐✈❡s ❡✐❣❡♥✈❛❧✉❡ ✶✱ S⊥

✶✷ ❣✐✈❡s ✵✱

■❢ λ,µ ∈ Spec(PS✶✷PS✷✸PS✶✷)\{✵,✶}✱ λ = µ t❤❡♥ ∄ ˆ R ❢♦r M ✳ Greatest ❞✐♠M✸ ⇐ ⇒ S✶✷ and S✷✸ are isoclinic modulo S✶✷ ∩S✷✸ ⇐ ⇒ ∃ ˆ R

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 38

✶✵✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❖rt❤♦❣♦♥❛❧ ❝❛s❡

❙♣❡❝✐❛❧ ❝❛s❡✿ V ✐s ❛ ✜♥✳ ❞✐♠✳ ❍✐❧❜❡rt s♣❛❝❡✱ T = S⊥✳ ❚❤❡ ❝♦♥✜❣✳ (S✶✷,S✷✸,S⊥

✶✷,S⊥ ✷✸) ✐s ❞❡t❡r♠✐♥❡❞ ❜②

Spec(ProjS✶✷ ◦ProjS✷✸ ◦ProjS✶✷) ❛s ❛ ♠✉❧t✐s❡t ✭◆❛♠❡ ✐♥ r❡❛❧ ❝❛s❡✿ ♣r✐♥❝✐♣❛❧ ❛♥❣❧❡s✳✮✳ ❋❡✇❡r ❞✐st✐♥❝t ❛♥❣❧❡s ⇒ ♠♦r❡ ❡♥❞♦♠♦r♣❤✐s♠s ♦❢ ρ✳ S✶✷ ∩S✷✸ ❣✐✈❡s ❡✐❣❡♥✈❛❧✉❡ ✶✱ S⊥

✶✷ ❣✐✈❡s ✵✱

■❢ λ,µ ∈ Spec(PS✶✷PS✷✸PS✶✷)\{✵,✶}✱ λ = µ t❤❡♥ ∄ ˆ R ❢♦r M ✳ Greatest ❞✐♠M✸ ⇐ ⇒ S✶✷ and S✷✸ are isoclinic modulo S✶✷ ∩S✷✸ ⇐ ⇒ ∃ ˆ R

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 39

✶✶✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

▼♦t✐✈❛t✐♦♥✴❛♣♣❧✐❝❛t✐♦♥

❆♣♣❧✐❝❛t✐♦♥✿ ❚❛❦❡✉❝❤✐✬s ❝♦♥❥❡❝t✉r❡✿ q✉❛♥t✉♠ ♦rt❤♦❣♦♥❛❧ ❜✐❛❧❣❡❜r❛ ˜ M+(✸) ։ Oq(so✸) ❤❛s ❛ P❇❲✲❜❛s✐s✳ ■t ✐s ❢❛❧s❡✦ ◆♦t❡s✿

t❤❡ ❍❡r♠✐t✐❛♥ s❝❛❧❛r ♣r♦❞✉❝t ✐s ♥♦t ♥❡❝❡ss❛r②✱ ❣❡♥❡r❛❧❧②✱ ♥♦♥✲s❡♠✐s✐♠♣❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ˜ D✹ ❛♣♣❡❛r✱ ❢✉rt❤❡r ❞✐r❡❝t✐♦♥s✿ r❡❧❛t❡❞ ❍♦♣❢ ❛❧❣❡❜r❛s✱ ❝♦r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ♦❢ M ✱ ❢✉rt❤❡r r✐♥❣ t❤❡♦r❡t✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ M ✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 40

✶✶✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

▼♦t✐✈❛t✐♦♥✴❛♣♣❧✐❝❛t✐♦♥

❆♣♣❧✐❝❛t✐♦♥✿ ❚❛❦❡✉❝❤✐✬s ❝♦♥❥❡❝t✉r❡✿ q✉❛♥t✉♠ ♦rt❤♦❣♦♥❛❧ ❜✐❛❧❣❡❜r❛ ˜ M+(✸) ։ Oq(so✸) ❤❛s ❛ P❇❲✲❜❛s✐s✳ ■t ✐s ❢❛❧s❡✦ ◆♦t❡s✿

t❤❡ ❍❡r♠✐t✐❛♥ s❝❛❧❛r ♣r♦❞✉❝t ✐s ♥♦t ♥❡❝❡ss❛r②✱ ❣❡♥❡r❛❧❧②✱ ♥♦♥✲s❡♠✐s✐♠♣❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ˜ D✹ ❛♣♣❡❛r✱ ❢✉rt❤❡r ❞✐r❡❝t✐♦♥s✿ r❡❧❛t❡❞ ❍♦♣❢ ❛❧❣❡❜r❛s✱ ❝♦r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ♦❢ M ✱ ❢✉rt❤❡r r✐♥❣ t❤❡♦r❡t✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ M ✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 41

✶✶✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

▼♦t✐✈❛t✐♦♥✴❛♣♣❧✐❝❛t✐♦♥

❆♣♣❧✐❝❛t✐♦♥✿ ❚❛❦❡✉❝❤✐✬s ❝♦♥❥❡❝t✉r❡✿ q✉❛♥t✉♠ ♦rt❤♦❣♦♥❛❧ ❜✐❛❧❣❡❜r❛ ˜ M+(✸) ։ Oq(so✸) ❤❛s ❛ P❇❲✲❜❛s✐s✳ ■t ✐s ❢❛❧s❡✦ ◆♦t❡s✿

t❤❡ ❍❡r♠✐t✐❛♥ s❝❛❧❛r ♣r♦❞✉❝t ✐s ♥♦t ♥❡❝❡ss❛r②✱ ❣❡♥❡r❛❧❧②✱ ♥♦♥✲s❡♠✐s✐♠♣❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ˜ D✹ ❛♣♣❡❛r✱ ❢✉rt❤❡r ❞✐r❡❝t✐♦♥s✿ r❡❧❛t❡❞ ❍♦♣❢ ❛❧❣❡❜r❛s✱ ❝♦r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ♦❢ M ✱ ❢✉rt❤❡r r✐♥❣ t❤❡♦r❡t✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ M ✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 42

✶✶✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

▼♦t✐✈❛t✐♦♥✴❛♣♣❧✐❝❛t✐♦♥

❆♣♣❧✐❝❛t✐♦♥✿ ❚❛❦❡✉❝❤✐✬s ❝♦♥❥❡❝t✉r❡✿ q✉❛♥t✉♠ ♦rt❤♦❣♦♥❛❧ ❜✐❛❧❣❡❜r❛ ˜ M+(✸) ։ Oq(so✸) ❤❛s ❛ P❇❲✲❜❛s✐s✳ ■t ✐s ❢❛❧s❡✦ ◆♦t❡s✿

t❤❡ ❍❡r♠✐t✐❛♥ s❝❛❧❛r ♣r♦❞✉❝t ✐s ♥♦t ♥❡❝❡ss❛r②✱ ❣❡♥❡r❛❧❧②✱ ♥♦♥✲s❡♠✐s✐♠♣❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ˜ D✹ ❛♣♣❡❛r✱ ❢✉rt❤❡r ❞✐r❡❝t✐♦♥s✿ r❡❧❛t❡❞ ❍♦♣❢ ❛❧❣❡❜r❛s✱ ❝♦r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ♦❢ M ✱ ❢✉rt❤❡r r✐♥❣ t❤❡♦r❡t✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ M ✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 43

✶✶✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

▼♦t✐✈❛t✐♦♥✴❛♣♣❧✐❝❛t✐♦♥

❆♣♣❧✐❝❛t✐♦♥✿ ❚❛❦❡✉❝❤✐✬s ❝♦♥❥❡❝t✉r❡✿ q✉❛♥t✉♠ ♦rt❤♦❣♦♥❛❧ ❜✐❛❧❣❡❜r❛ ˜ M+(✸) ։ Oq(so✸) ❤❛s ❛ P❇❲✲❜❛s✐s✳ ■t ✐s ❢❛❧s❡✦ ◆♦t❡s✿

t❤❡ ❍❡r♠✐t✐❛♥ s❝❛❧❛r ♣r♦❞✉❝t ✐s ♥♦t ♥❡❝❡ss❛r②✱ ❣❡♥❡r❛❧❧②✱ ♥♦♥✲s❡♠✐s✐♠♣❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ˜ D✹ ❛♣♣❡❛r✱ ❢✉rt❤❡r ❞✐r❡❝t✐♦♥s✿ r❡❧❛t❡❞ ❍♦♣❢ ❛❧❣❡❜r❛s✱ ❝♦r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ♦❢ M ✱ ❢✉rt❤❡r r✐♥❣ t❤❡♦r❡t✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ M ✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 44

✶✷✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s

SLIDE 45

✶✸✴✶✸ ◗✉❛♥t✉♠ ▼❛tr✐❝❡s ❯♥✐✈❡rs❛❧ ❜✐❛❧❣❡❜r❛s ❉❡✜♥✐t✐♦♥ Pr♦♣❡rt✐❡s ■❞❡❛ ♦❢ t❤❡ ♣r♦♦❢

❘❡❢❡r❡♥❝❡s

▼✳ ❆rt✐♥✱ ❲✳ ❙❝❤❡❧t❡r✱ ❏✳ ❚❛t❡✱ ◗✉❛♥t✉♠ ❞❡❢♦r♠❛t✐♦♥s ♦❢ ●▲✱ ❈♦♠♠✉♥✳ P✉r❡ ❆♣♣t✳▼❛t❤✳ ✹✹ ✭✶✾✾✶✮✱ ✽✼✾✲✽✾✺✳ ❑✳ ❆✳ ❇r♦✇♥✱ ❑✳ ❘✳ ●♦♦❞❡❛r❧✱ ▲❡❝t✉r❡s ♦♥ ❆❧❣❡❜r❛✐❝ ◗✉❛♥t✉♠

r♦✉♣s✱ ❇✐r❦❤❛✉s❡r✱ ✷✵✵✷✳

❆✳❲✳ ❈❤❛tt❡rs✱ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ ✉♥✐q✉❡ ❢❛❝t♦r✐s❛t✐♦♥ ❞♦♠❛✐♥s✱ ▼❛t❤✳ Pr♦❝✳ ❈❛♠❜r✐❞❣❡ P❤✐❧✳ ❙♦❝✳ ✾✺ ✭✶✾✽✹✮✱ ✹✾✕✺✹✳ ▼✳ ❉♦♠♦❦♦s✱ ❚✳ ❍✳ ▲❡♥❛❣❛♥✱ ❈♦♥❥✉❣❛t✐♦♥ ❝♦✐♥✈❛r✐❛♥ts ♦❢ q✉❛♥t✉♠ ♠❛tr✐❝❡s✱ ❇✉❧❧✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳ ✸✺ ✭✷✵✵✸✮✳ ▼✳ ❉♦♠♦❦♦s✱ ❚✳ ❍✳ ▲❡♥❛❣❛♥✱ ❘❡♣r❡s❡♥t❛t✐♦♥s r✐♥❣s ♦❢ q✉❛♥t✉♠ ❣r♦✉♣s✱ ❏✳ ❆❧❣❡❜r❛ ✷✽✷ ✭✷✵✵✹✮✳ ❚✳ ❍❛②❛s❤✐✱ ◗✉❛♥t✉♠ ❞❡❢♦r♠❛t✐♦♥s ♦❢ ❝❧❛ss✐❝❛❧ ❣r♦✉♣s✱ P✉❜❧✳ ❘■▼❙ ❑②♦t♦ ❯♥✐✈✳ ✷✽ ✭✶✾✾✷✮✱ ✺✼✲✽✶✳ ❙③✳ ▼és③ár♦s✱ ❈♦❝♦♠♠✉t❛t✐✈❡ ❡❧❡♠❡♥ts ❢♦r♠ ❛ ♠❛①✐♠❛❧ ❝♦♠♠✉t❛t✐✈❡ s✉❜❛❧❣❡❜r❛ ✐♥ q✉❛♥t✉♠ ♠❛tr✐❝❡s✱ s✉❜♠✐tt❡❞✳ ❆✳ ❙✉❞❜❡r②✱ ▼❛tr✐①✲❊❧❡♠❡♥t ❇✐❛❧❣❡❜r❛s ❉❡t❡r♠✐♥❡❞ ❜② ◗✉❛❞r❛t✐❝ ❈♦♦r❞✐♥❛t❡ ❆❧❣❡❜r❛s✱ ❏✳ ❆❧❣✳ ✶✺✽ ✭✶✾✾✸✮✱ ♥♦✳ ✷✱ ✸✼✺✲✸✾✾✳

❙③❛❜♦❧❝s ▼és③ár♦s ◗✉❛♥t✐③❡❞ ❈♦♦r❞✐♥❛t❡ ❘✐♥❣s ❛♥❞ ❯♥✐✈❡rs❛❧ ❇✐❛❧❣❡❜r❛s