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Parity of the number of boxes of the gray parts depends on the parity of the height j. 

Parity of the number of boxes of the gray parts depends on the parity of the height j. 

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Figure 3. Number of boxes of the gray parts is even. 
Figure 4. Parity of the number of boxes of the gray parts depends on...
Figure 5. Partition µ for the parts k n k (k is odd). 
Figure 6. Partition µ for the parts k n k ( k is even).
+12 Figure 7. One cycle of partition µ between successive odd number n j...
Fingerprint Invariant of Partitions and Construction
Article
Full-text available
  • Nov 2017
The fingerprint invariant of partitions can be used to describe the Kazhdan-Lusztig map for the classical groups. We discuss the basic properties of fingerprint. We construct the fingerprints of rigid partitions in the $B_n$, $C_n$, and $D_n$ theories. To calculate the fingerprint of a rigid semisimple operator $(\lambda^{'};\lambda^{"})$, we decom...
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Figure 4. The images of III type blocks under the map µ. The II type...
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Invariants of partitions
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  • Nov 2017
The symbol invariant is used to describe the Springer correspondence for the classical groups by Lusztig. And the fingerprint invariant can be used to describe the Kazhdan-Lusztig map. They are invariants of rigid semisimple operators labeled by pairs of partitions $(\lambda^{'}, \lambda^{"})$. It is conjectured that the symbol invariant is equival...
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... Another construction of symbol is given in [17]. We discuss the basic properties of fingerprint and its constructions in [18]. ...
... Then we present the construction of the representative element (λ ′ , λ ′′ ) R of rigid surface operators with the given symbol. In Section 4, we give the constructions of the fingerprint invariant in [18]. We calculate the fingerprint of the representative element (λ ′ , λ ′′ ) R . ...
... In the first two subsections, we would like to introduce the construction of the fingerprint in [18]. Then we try to find a representative element of the rigid semisimple operators with the same fingerprint and calculate symbol of it. ...
Invariants of partitions and representative elements
Article
Full-text available
  • Nov 2017
The symbol invariant is used to describe the Springer correspondence for the classical groups by Lusztig. And the fingerprint invariant can be used to describe the Kazhdan-Lusztig map. They are invariants of rigid semisimple operators described by pairs of partitions $(\lambda^{'}, \lambda^{"})$. We construct a nice representative element of the rigid semisimple operators with the same symbol invariant. The fingerprint of the representative element can be obtained immediately. We also discuss the representative element of rigid semisimple operator with the same fingerprint invariant. Our construction can be regarded as the maps between these two invariants.
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