PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2>2bb2M#2`; p`B2iB2b>Qx?B w2M;
>mx?QM; lMBp2`bBiv Q7 a+B2M+2 M/ h2+?MQHQ;v
CQBMi rQ`F rBi? >B`Fm #2 M/ LQFB 6mDBi

J`+? k9- kykR

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2bh#H2 Q7 +QMi2Mib

6mHH 6H; p`B2iv>2bb2M#2`; p`B2iB2b
6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2bq2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b6mHH 6H; p`B2iv

h?2 /2}MBiBQM Q7 7mHH ~; p`B2iv
h?2 7mHH ~; p`B2iv Flag(CM ) +QMbBbib Q7 M2bi2/ b2[m2M+2b Q7
HBM2` bm#bT+2b Q7 CMo• = (oR ⊂ ok ⊂ · · · ⊂ oM = CM )r?2`2 oB Bb Q7 +QKTH2t /BK2MbBQM BX

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b6mHH 6H; p`B2iv

h?2 /2}MBiBQM Q7 7mHH ~; p`B2iv
h?2 7mHH ~; p`B2iv Flag(CM ) +QMbBbib Q7 M2bi2/ b2[m2M+2b Q7
HBM2` bm#bT+2b Q7 CMo• = (oR ⊂ ok ⊂ · · · ⊂ oM = CM )r?2`2 oB Bb Q7 +QKTH2t /BK2MbBQM BX:GM (C) +ib QM Flag(CM ) i`MbBiBp2Hv M/ i?2 BbQi`QTv ;`QmT" Q7 i?2 TQBMiy ⊂ (2R ) ⊂ (2R , 2k ) ⊂ · · · ⊂ (2R , 2k , · · · , 2M )+QMbBbib Q7 mTT2` i`BM;mH` BMp2`iB#H2 Ki`B+2bX >2M+2
Flag(CM ) Bb  MQMbBM;mH` T`QD2+iBp2 p`B2ivX

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b6mHH 6H; p`B2iv

hQ`mb +iBQM QM Flag(CM ):GM (C)/" ∼= Flag(CM ).
;" #→ y ⊆ &;R ' ⊆ &;R , ;k ' ⊆ · · · ⊆ &;R , ;k , · · · , ;M '.

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b6mHH 6H; p`B2iv

hQ`mb +iBQM QM Flag(CM ):GM (C)/" ∼= Flag(CM ).
;" #→ y ⊆ &;R ' ⊆ &;R , ;k ' ⊆ · · · ⊆ &;R , ;k , · · · , ;M '.
G2i h #2 i?2 ;`QmT +QMbBbiBM; Q7 i?2 BMp2`iB#H2 M/ /B;QMH
Ki`B+2bX h?2 H27i +iBQM h ! :GM (C) BM/m+2b  h +iBQM QM:GM (C)/".

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b6mHH 6H; p`B2iv

hQ`mb +iBQM QM Flag(CM ):GM (C)/" ∼= Flag(CM ).
;" #→ y ⊆ &;R ' ⊆ &;R , ;k ' ⊆ · · · ⊆ &;R , ;k , · · · , ;M '.
G2i h #2 i?2 ;`QmT +QMbBbiBM; Q7 i?2 BMp2`iB#H2 M/ /B;QMH
Ki`B+2bX h?2 H27i +iBQM h ! :GM (C) BM/m+2b  h +iBQM QM:GM (C)/".
Flag(CM )h ∼= SM .

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b>2bb2M#2`; 7mM+iBQMb

G2i M #2  TQbBiBp2 BMi2;2`- M/ r2 b2i [M] := {R, k, · · · , M}.

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b>2bb2M#2`; 7mM+iBQMb

G2i M #2  TQbBiBp2 BMi2;2`- M/ r2 b2i [M] := {R, k, · · · , M}.

.2}MBiBQM

 >2bb2M#2`; 7mM+iBQM Bb  7mM+iBQM ? : [M] → [M] r?B+? biBb}2bi?2 7QHHQrBM; +QM/BiBQMb,

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b>2bb2M#2`; 7mM+iBQMb

G2i M #2  TQbBiBp2 BMi2;2`- M/ r2 b2i [M] := {R, k, · · · , M}.

.2}MBiBQM

 >2bb2M#2`; 7mM+iBQM Bb  7mM+iBQM ? : [M] → [M] r?B+? biBb}2bi?2 7QHHQrBM; +QM/BiBQMb,
R?(B + R) ≥ ?(B) 7Q` HH R ≤ B ≤ M − R-

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b>2bb2M#2`; 7mM+iBQMb

G2i M #2  TQbBiBp2 BMi2;2`- M/ r2 b2i [M] := {R, k, · · · , M}.

.2}MBiBQM

 >2bb2M#2`; 7mM+iBQM Bb  7mM+iBQM ? : [M] → [M] r?B+? biBb}2bi?2 7QHHQrBM; +QM/BiBQMb,
Rk?(B + R) ≥ ?(B) 7Q` HH R ≤ B ≤ M − R?(B) ≥ B 7Q` HH R ≤ B ≤ MX

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b1tKTH2b

q2 +M 2tT`2bb  >2bb2M#2`; 7mM+iBQM #v HBbiBM; Bib pHm2b BM 
b2[m2M+2 b ? = (?(R), ?(k), · · · , ?(M)).

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b1tKTH2b

q2 +M 2tT`2bb  >2bb2M#2`; 7mM+iBQM #v HBbiBM; Bib pHm2b BM 
b2[m2M+2 b ? = (?(R), ?(k), · · · , ?(M)).

1tKTH2R

M = 8 M/ ? = (k, j, 9, 8, 8)c

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b1tKTH2b

q2 +M 2tT`2bb  >2bb2M#2`; 7mM+iBQM #v HBbiBM; Bib pHm2b BM 
b2[m2M+2 b ? = (?(R), ?(k), · · · , ?(M)).

1tKTH2R

M = 8 M/ ? = (k, j, 9, 8, 8)ck

M = 8 M/ ? = (8, 8, 8, 8, 8).

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b1tT`2bb >2bb2M#2`; 7mM+iBQM #v biB`+b2 /B;`KR

M = 8 M/ ? = (k, j, 9, 8, 8)XPM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b1tT`2bb >2bb2M#2`; 7mM+iBQM #v biB`+b2 /B;`KR

M = 8 M/ ? = (k, j, 9, 8, 8)Xk

M = 8 M/ ? = (8, 8, 8, 8, 8)XPM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b

S`iBH Q`/2` QM >2bb2M#2`; 7mM+iBQMb

G2i ?, ?! : [M] → [M] #2 >2bb2M#2`; 7mM+iBQMbX A7 7Q` Mv B ∈ [M]?(B) ≤ ?! (B) i?2M r2 bv ? , ?! .

1tKTH2""•

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b.mH >2bb2M#2`; 7mM+iBQMb
6Q`  >2bb2M#2`; 7mM+iBQM ? : [M] → [M]- r2 +M /2}M2  M2r
>2bb2M#2`; 7mM+iBQM ?∗ : [M] → [M] b 7QHHQrbX?∗ (B) = |{D ∈ [M]|?(D) ≥ M + R − B}|.

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b.mH >2bb2M#2`; 7mM+iBQMb
6Q`  >2bb2M#2`; 7mM+iBQM ? : [M] → [M]- r2 +M /2}M2  M2r
>2bb2M#2`; 7mM+iBQM ?∗ : [M] → [M] b 7QHHQrbX?∗ (B) = |{D ∈ [M]|?(D) ≥ M + R − B}|.

1tKTH2

G2i M = 8- ? = (j, j, 9, 8, 8) i?2M ?∗ = (k, j, 8, 8, 8).

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b.mH >2bb2M#2`; 7mM+iBQMb
6Q`  >2bb2M#2`; 7mM+iBQM ? : [M] → [M]- r2 +M /2}M2  M2r
>2bb2M#2`; 7mM+iBQM ?∗ : [M] → [M] b 7QHHQrbX?∗ (B) = |{D ∈ [M]|?(D) ≥ M + R − B}|.

1tKTH2

G2i M = 8- ? = (j, j, 9, 8, 8) i?2M ?∗ = (k, j, 8, 8, 8).
!

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b_2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b

G2i a #2 M M × M `2;mH` b2KBbBKTH2 Ki`Bt UBX2X M M × M/B;QMHBx#H2 Ki`Bt rBi? M /BbiBM+i 2B;2MpHm2bV M/ ? : [M] → [M]#2  >2bb2M#2`; 7mM+iBQM- r2 +M /2}M2  bm#p`B2iv Q7 i?2 7mHH
~; p`B2iv Flag(CM ) b 7QHHQrbX
Hess(a, ?) := {o• ∈ Flag(CM )|aoB ⊆ o?(B) }.
Hess(a, ?) Bb +HH2/  `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2ivX

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b

bbmKTiBQM M/ MQiiBQM

h?2 ;2M2`H HBM2` ;`QmT :GM (C) H27i +ib QM Flag(CM ) bQ r2?p2 Hess(;a;−R , ?) = ; Hess(a, ?) BM Flag(CM ) 7Q` HH; ∈ :G(CM )X Ai 7QHHQrb i?i Hess(;a;−R , ?) ∼= Hess(a, ?) 7Q`
HH ; ∈ :G(CM )X >2M+2- r2 +M Hrvb bbmK2 i?i i?2`2;mH` b2KBbBKTH2 Ki`Bt a Bb  /B;QMH Ki`BtX

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b2tKTH2b

1tKTH2R

A7 ?(B) = M 7Q` HH B ∈ [M]- i?2M 7Q` Mv a r2 ?p2
Hess(a, ?) = Flag(CM )XPM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b2tKTH2b

1tKTH2Rk

A7 ?(B) = M 7Q` HH B ∈ [M]- i?2M 7Q` Mv a r2 ?p2
Hess(a, ?) = Flag(CM )X
A7 ?(B) = B + R 7Q` B = R, k, · · · , M − R- i?2M Hess(a, ?) Bb +HH2/i?2 T2`KmiQ?2/`H p`B2iv r?B+? Bb i?2 bKQQi? T`QD2+iBp2 iQ`B+
p`B2iv +Q``2bTQM/BM; iQ i?2 7M +QMbBbiBM; Q7 i?2 +QHH2+iBQM Q7
q2vH +?K#2`b BM ivT2 M−R X

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b

S`QT2`iB2bh?2 7QHHQrBM; T`QT2`iB2b QM `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b+M #2 7QmM/ BM .2 J`B- S`Q+2bB M/ a?vKMǶb MB+2 TT2` (9)X
R

Hess(a, ?) Bb bKQQi?cPM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b

S`QT2`iB2bh?2 7QHHQrBM; T`QT2`iB2b QM `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b+M #2 7QmM/ BM .2 J`B- S`Q+2bB M/ a?vKMǶb MB+2 TT2` (9)XR
k

Hess(a, ?) Bb bKQQi?c!
dimC MB=R (?(B) − B)XPM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b

hQ`mb +iBQM QM Hess(a, ?)G2i h #2 i?2 7QHHQrBM; +QKTH2t iQ`mb +QMbBbiBM; Q7 /B;QMH
Ki`B+2b,
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h?2M h +ib QM Flag(CM ) pB i?2 :GM (C)@+iBQM QM Flag(CM )X
aBM+2 i?2 Ki`Bt a /2}MBM; Hess(a, ?) Bb /B;QMH- HH i?2 2H2K2Mib
Q7 h +QKKmi2 rBi? aX h?2`27Q`2- i?2 h@+iBQM QM Flag(CM )T`2b2`p2b Hess(a, ?)- M/ ?2M+2 h +ib QM Hess(a, ?)XPM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2bh?2 }t2/ TQBMib b2i Hess(a, ?)h_2+HH i?i r2 ?p2 Flag(CM )h = SM #v B/2MiB7vBM; r ∈ SM
M/ o• - r?2`2 oB = C2r(R) ⊕ C2r(k) ⊕ · · · ⊕ C2r(B) 7Q` HH
R ≤ B ≤ MX AM 7+i- Hess(a, ?) +QMiBMb HH i?2 h@}t2/ TQBMib
Q7 Flag(CM )h ,Hess(a, ?)h = Flag(CM )h = SM .

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2bh?2 }t2/ TQBMib b2i Hess(a, ?)h_2+HH i?i r2 ?p2 Flag(CM )h = SM #v B/2MiB7vBM; r ∈ SM
M/ o• - r?2`2 oB = C2r(R) ⊕ C2r(k) ⊕ · · · ⊕ C2r(B) 7Q` HH
R ≤ B ≤ MX AM 7+i- Hess(a, ?) +QMiBMb HH i?2 h@}t2/ TQBMib
Q7 Flag(CM )h ,Hess(a, ?)h = Flag(CM )h = SM .
S`Q#H2K A7 s Bb  h@BMp`BMi MQMbBM;mH` bm#p`B2iv Q7 Flag(CM ) bm+?
i?i χ(s) = χ(Flag(CM ))- i?2M b?QmH/ s #2 BbQKQ`T?BbK iQ
bQK2 Hess(a, ?)\PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b_2K`F_2+2MiHv- vx2M#2`; M/ "m+?bi#2` +QMbi`m+i2/  7KBHv Q7
bKQQi? bm#KMB7QH/b s(?)! b Q7 Flag(CM ) rBi? +QKT+i iQ`mb T
+iBQM ?p2 i?2 7QHHQrBM; T`QT2`iB2b,

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b_2K`F_2+2MiHv- vx2M#2`; M/ "m+?bi#2` +QMbi`m+i2/  7KBHv Q7
bKQQi? bm#KMB7QH/b s(?)! b Q7 Flag(CM ) rBi? +QKT+i iQ`mb T
+iBQM ?p2 i?2 7QHHQrBM; T`QT2`iB2b,∼ >∗ (Hess(a, ?));
R > (s(?)) =∗PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b>2bb2M#2`; p`B2iB2b_2K`F_2+2MiHv- vx2M#2`; M/ "m+?bi#2` +QMbi`m+i2/  7KBHv Q7
bKQQi? bm#KMB7QH/b s(?)! b Q7 Flag(CM ) rBi? +QKT+i iQ`mb T
+iBQM ?p2 i?2 7QHHQrBM; T`QT2`iB2b,∼ >∗ (Hess(a, ?));
R > (s(?)) =∗ks(?)T ∼= Hess(a, ?)h .

.2iBHb +M #2 7QmM/ BM (k)XPM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b
6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2bq2B;?ibR

q2B;?i, M H;2#`B+ ?QKQKQ`T?BbK µ : h → C∗ . >2M+2{q2B;?ib} ∼= ZM .

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b
6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2bq2B;?ibR

q2B;?i, M H;2#`B+ ?QKQKQ`T?BbK µ : h → C∗ . >2M+2{q2B;?ib} ∼= ZM .

k.QKBMMi r2B;?ib,{µ = (µR , µk , · · · , µM ) ∈ ZM |µR ≥ µk ≥ · · · ≥ µM }

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b
6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2bq2B;?ibR

q2B;?i, M H;2#`B+ ?QKQKQ`T?BbK µ : h → C∗ . >2M+2{q2B;?ib} ∼= ZM .

k.QKBMMi r2B;?ib,{µ = (µR , µk , · · · , µM ) ∈ ZM |µR ≥ µk ≥ · · · ≥ µM }j

_2;mH` /QKBMMi r2B;?ib,{µ = (µR , µk , · · · , µM ) ∈ ZM |µR > µk > · · · > µM }

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b
6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b

GBM2 #mM/H2b bbQ+Bi2/ iQ r2B;?ibCµ #2 i?2 QM2 /BK2MbBQMH +QKTH2t `2T`2b2MiiBQM Q7 h rBi?
r2B;?i µ = (µR , µk , · · · , µM ). *QMbB/2` i?2 T`QD2+iBQM " " hi?2M Cµ Bb HbQ "@KQ/mH2 ;Bp2M #v
#R ∗ ∗ ∗#k ∗ ∗ µ µ · p = #R R #k k · · · #µM M p.
X
X

X ∗
#M

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b
6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b

GBM2 #mM/H2b bbQ+Bi2/ iQ r2B;?ibCµ #2 i?2 QM2 /BK2MbBQMH +QKTH2t `2T`2b2MiiBQM Q7 h rBi?
r2B;?i µ = (µR , µk , · · · , µM ). *QMbB/2` i?2 T`QD2+iBQM " " hi?2M Cµ Bb HbQ "@KQ/mH2 ;Bp2M #v
#R ∗ ∗ ∗#k ∗ ∗ µ µ · p = #R R #k k · · · #µM M p.
X
X

X ∗
#M

Gµ := : ×" C∗µ - i?2 "@+iBQM QM : × *∗µ Bb ;Bp2M #v(;, p)# := (;#, #−R · p) = (;#, µ(#)p).
q2 HbQ /2MQi2 Gµ |Hess(a,?) #v Gµ .

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b
6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b

KTH2 M/ M27 HBM2 #mM/H2b
S`QTQbBiBQM U(j)VR

Gµ Bb KTH2 Qp2` FlagM (C) B7 M/ QMHv B7 µ Bb  `2;mH`/QKBMMi r2B;?iX

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b
6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b

KTH2 M/ M27 HBM2 #mM/H2b
S`QTQbBiBQM U(j)VRk

Gµ Bb KTH2 Qp2` FlagM (C) B7 M/ QMHv B7 µ Bb  `2;mH`/QKBMMi r2B;?iX
Gµ Bb M27 Qp2` FlagM (C) B7 M/ QMHv B7 µ Bb  /QKBMMi r2B;?iX*Q`QHH`v

bbmK2 i?i ?(B) ≥ B + R 7Q` R ≤ B ≤ M − R- i?2M
R

Gµ Bb KTH2 Qp2` Hess(a, ?) B7 M/ QMHv B7 µ Bb  `2;mH`/QKBMMi r2B;?iX

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b
6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b

KTH2 M/ M27 HBM2 #mM/H2b
S`QTQbBiBQM U(j)VRk

Gµ Bb KTH2 Qp2` FlagM (C) B7 M/ QMHv B7 µ Bb  `2;mH`/QKBMMi r2B;?iX
Gµ Bb M27 Qp2` FlagM (C) B7 M/ QMHv B7 µ Bb  /QKBMMi r2B;?iX*Q`QHH`v

bbmK2 i?i ?(B) ≥ B + R 7Q` R ≤ B ≤ M − R- i?2MRk

Gµ Bb KTH2 Qp2` Hess(a, ?) B7 M/ QMHv B7 µ Bb  `2;mH`/QKBMMi r2B;?iX
Gµ Bb M27 Qp2` Hess(a, ?) B7 M/ QMHv B7 µ Bb  /QKBMMi r2B;?iX

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b
6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2bh?2 MiB@+QMQMB+H #mM/H2

S`QTQbBiBQM
O(−EHess(a,?) ) ∼
= Gξ?

r?2`2
ξ? =

0

R≤B<D≤?(B)(iB − iD ) M/ iB = (y, y, · · · , R, y, · · · , y)

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b
6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b1tKTH2
1tKTH2

A7 ? = (k, j, 9, 8, 8) i?2M ξ? = iR − i8 .

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b
6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2bq?2M Hess(a, ?) Bb 6MQ

6MQ, A7 O(−EHess(a,?) ) Bb KTH2X

h?2Q`2K U#2@6mDBi@wV

bbmK2 i?i ?(B) ≥ B + R 7Q` HH R ≤ B ≤ M − RX h?2M Hess(a, ?) Bb
6MQ B7 M/ QMHv B7? = (F + R, F + k, · · · , M, M, · · · , M)
bm+? i?i M−R
k ≤ F ≤ M − R.

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b
6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b1tKTH2

1tKTH2R

A7 ? = (k, j, j)- Hess(a, ?) Bb 6MQXk

A7 ? = (k, j, 9, 9)- Hess(a, ?) Bb MQi 6MQX

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2bq2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b.2}MBiBQM

q2F 6MQ, A7 O(−EHess(a,?) ) Bb M27 M/ #B;XS`QTQbBiBQM U(8)V

A7 O(−EHess(a,?) ) Bb M27 i?2M i?2 7QHHQrBM; +QM/BiBQMb `2
2[mBpH2Mi,RkO(−EHess(a,?) ) Bb #B;c1/
Hess(a,?) +R (O(−EHess(a,?) )) > y r?2`2 / = dimC Hess(a, ?).

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2bq2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2bq?2M Hess(a, ?) Bb r2F 6MQ

h?2Q`2K U#2@6mDBi@wV
bbmK2 i?i ?(B) ≥ B + R 7Q` HH R ≤ B ≤ M − RX h?2M Hess(a, ?) Bb
r2F 6MQ B7 M/ QMHv B7?(B) − ?(B + R) + k − ?∗ (M + R − B) + ?∗ (M − B) ≥ y
7Q` HH R ≤ B ≤ M − R.

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2bq2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b1tKTH2b

1tKTH2R

A7 ? = (k, j, 9, 9)- Hess(a, ?) Bb r2F 6MQX

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2bq2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b1tKTH2b

1tKTH2R

A7 ? = (k, j, 9, 9)- Hess(a, ?) Bb r2F 6MQXk

A7 ? = (j, j, 9, 9)- Hess(a, ?) Bb r2F 6MQX

PM 6MQ M/ r2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2bq2F 6MQ `2;mH` b2KBbBKTH2 >2bb2M#2`; p`B2iB2b`272`2M+2b>X #2- LX 6mDBi M/ >X w2M;- 6MQ M/ r2F 6MQ
>2bb2M#2`; p`B2iB2b- `sBp, kyyjXRkk3epRX
X vx2M#2`; M/ oX "m+?bi#2`- JMB7QH/b Q7 BbQbT2+i`H
Ki`B+2b M/ >2bb2M#2`; p`B2iB2b- `sBp, R3yjXyRRjkpkX
JX "`BQM- G2+im`2b QM i?2 ;2QK2i`v Q7 ~; p`B2iB2b- BM hQTB+b
BM *Q?QKQHQ;B+H aim/B2b Q7 H;2#`B+ o`B2iB2b- h`2M/b Ji?X"B`F? mb2`- "b2H- kyy8- jjĜ38X
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