What does the necklace symbolize when Mathilde first sees it in the satin box in "The Necklace"?

The necklace symbolizes the unattainable life of wealth that Mathilde longs for.

Mathilde is a middle class woman who longs to be rich.  She feels like she was born into the wrong lifestyle.  She is pretty, but she wants wealth.  When her husband gets her an invitation to a party, she is not as happy as he might have thought.  She gets very upset, and he asks her what is wrong.

"Nothing. Only I have...

The necklace symbolizes the unattainable life of wealth that Mathilde longs for.

Mathilde is a middle class woman who longs to be rich.  She feels like she was born into the wrong lifestyle.  She is pretty, but she wants wealth.  When her husband gets her an invitation to a party, she is not as happy as he might have thought.  She gets very upset, and he asks her what is wrong.

"Nothing. Only I have no gown, and, therefore, I can't go to this ball. Give your card to some colleague whose wife is better equipped than I am."

Mathilde convinces her husband to use all of his savings on the new dress. However, she still needs a jewel that can compete with the other wealthy ball attendees.  He tells her to borrow one from a rich friend.

Madame Forestier is an old school friend of Mathilde’s, and much wealthier.  She gladly agrees to lend Mathilde any jewel that she wants.  She offers up many to her, but Mathilde chooses one that seems particularly expensive.

Suddenly she discovered, in a black satin box, a superb diamond necklace, and her heart throbbed with an immoderate desire. Her hands trembled as she took it. She fastened it round her throat,outside her high-necked waist, and was lost in ecstasy at her reflection in the mirror.

The necklace seems very valuable, and appears to be part of the lifestyle Mathilde desires and feels that she deserves.  It symbolizes wealth and prestige to Mathilde.  She has no idea that it’s fake.  When she loses it, she replaces it with the real thing.  Madame Forestier doesn’t know it until years later, when Mathilde has paid off the debt.  Mathilde tells her, and she realizes that she has wasted her life and her beauty paying off a debt that she didn’t need to incur.